mathlib documentation

topology.subset_properties

Properties of subsets of topological spaces #

In this file we define various properties of subsets of a topological space, and some classes on topological spaces.

Main definitions #

We define the following properties for sets in a topological space:

For each of these definitions (except for is_clopen), we also have a class stating that the whole space satisfies that property: compact_space, irreducible_space

Furthermore, we have three more classes:

On the definition of irreducible and connected sets/spaces #

In informal mathematics, irreducible spaces are assumed to be nonempty. We formalise the predicate without that assumption as is_preirreducible. In other words, the only difference is whether the empty space counts as irreducible. There are good reasons to consider the empty space to be “too simple to be simple” See also https://ncatlab.org/nlab/show/too+simple+to+be+simple, and in particular https://ncatlab.org/nlab/show/too+simple+to+be+simple#relationship_to_biased_definitions.

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def is_compact {α : Type u} [topological_space α] (s : set α) :
Prop

A set s is compact if for every nontrivial filter f that contains s, there exists a ∈ s such that every set of f meets every neighborhood of a.

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theorem is_compact.compl_mem_sets {α : Type u} [topological_space α] {s : set α} (hs : is_compact s) {f : filter α} (hf : ∀ (a : α), a ∈ s → sᶜ ∈ 𝓝 a ⊓ f) :
sᶜ ∈ f

The complement to a compact set belongs to a filter f if it belongs to each filter 𝓝 a ⊓ f, a ∈ s.

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theorem is_compact.compl_mem_sets_of_nhds_within {α : Type u} [topological_space α] {s : set α} (hs : is_compact s) {f : filter α} (hf : ∀ (a : α), a ∈ s → (∃ (t : set α) (H : t ∈ 𝓝[s] a), tᶜ ∈ f)) :
sᶜ ∈ f

The complement to a compact set belongs to a filter f if each a ∈ s has a neighborhood t within s such that tᶜ belongs to f.

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theorem is_compact.induction_on {α : Type u} [topological_space α] {s : set α} (hs : is_compact s) {p : set α → Prop} (he : p ∅) (hmono : ∀ ⦃s t : set α⦄, s ⊆ t → p t → p s) (hunion : ∀ ⦃s t : set α⦄, p s → p t → p (s ∪ t)) (hnhds : ∀ (x : α), x ∈ s → (∃ (t : set α) (H : t ∈ 𝓝[s] x), p t)) :
p s

If p : set α → Prop is stable under restriction and union, and each point x of a compact set s has a neighborhood t within s such that p t, then p s holds.

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theorem is_compact.inter_right {α : Type u} [topological_space α] {s t : set α} (hs : is_compact s) (ht : is_closed t) :
is_compact (s ∩ t)

The intersection of a compact set and a closed set is a compact set.

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theorem is_compact.inter_left {α : Type u} [topological_space α] {s t : set α} (ht : is_compact t) (hs : is_closed s) :
is_compact (s ∩ t)

The intersection of a closed set and a compact set is a compact set.

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theorem is_compact.diff {α : Type u} [topological_space α] {s t : set α} (hs : is_compact s) (ht : is_open t) :
is_compact (s \ t)

The set difference of a compact set and an open set is a compact set.

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theorem compact_of_is_closed_subset {α : Type u} [topological_space α] {s t : set α} (hs : is_compact s) (ht : is_closed t) (h : t ⊆ s) :
is_compact t

A closed subset of a compact set is a compact set.

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theorem is_compact.adherence_nhdset {α : Type u} [topological_space α] {s t : set α} {f : filter α} (hs : is_compact s) (hf₂ : f ≤ 𝓟 s) (ht₁ : is_open t) (ht₂ : ∀ (a : α), a ∈ s → cluster_pt a f → a ∈ t) :
t ∈ f
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theorem is_compact_iff_ultrafilter_le_nhds {α : Type u} [topological_space α] {s : set α} :
is_compact s ↔ ∀ (f : ultrafilter α), ↑f ≤ 𝓟 s → (∃ (a : α) (H : a ∈ s), ↑f ≤ 𝓝 a)
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theorem is_compact.ultrafilter_le_nhds {α : Type u} [topological_space α] {s : set α} :
is_compact s → ∀ (f : ultrafilter α), ↑f ≤ 𝓟 s → (∃ (a : α) (H : a ∈ s), ↑f ≤ 𝓝 a)

Alias of is_compact_iff_ultrafilter_le_nhds.

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theorem is_compact.elim_directed_cover {α : Type u} [topological_space α] {s : set α} {ι : Type v} [hι : nonempty ι] (hs : is_compact s) (U : ι → set α) (hUo : ∀ (i : ι), is_open (U i)) (hsU : s ⊆ ⋃ (i : ι), U i) (hdU : directed has_subset.subset U) :
∃ (i : ι), s ⊆ U i

For every open directed cover of a compact set, there exists a single element of the cover which itself includes the set.

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theorem is_compact.elim_finite_subcover {α : Type u} [topological_space α] {s : set α} {ι : Type v} (hs : is_compact s) (U : ι → set α) (hUo : ∀ (i : ι), is_open (U i)) (hsU : s ⊆ ⋃ (i : ι), U i) :
∃ (t : finset ι), s ⊆ ⋃ (i : ι) (H : i ∈ t), U i

For every open cover of a compact set, there exists a finite subcover.

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theorem is_compact.elim_nhds_subcover' {α : Type u} [topological_space α] {s : set α} (hs : is_compact s) (U : Π (x : α), x ∈ s → set α) (hU : ∀ (x : α) (H : x ∈ s), U x H ∈ 𝓝 x) :
∃ (t : finset ↥s), s ⊆ ⋃ (x : ↥s) (H : x ∈ t), U ↑x _
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theorem is_compact.elim_nhds_subcover {α : Type u} [topological_space α] {s : set α} (hs : is_compact s) (U : α → set α) (hU : ∀ (x : α), x ∈ s → U x ∈ 𝓝 x) :
∃ (t : finset α), (∀ (x : α), x ∈ t → x ∈ s) ∧ s ⊆ ⋃ (x : α) (H : x ∈ t), U x
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theorem is_compact.elim_finite_subfamily_closed {α : Type u} [topological_space α] {s : set α} {ι : Type v} (hs : is_compact s) (Z : ι → set α) (hZc : ∀ (i : ι), is_closed (Z i)) (hsZ : (s ∩ ⋂ (i : ι), Z i) = ∅) :
∃ (t : finset ι), (s ∩ ⋂ (i : ι) (H : i ∈ t), Z i) = ∅

For every family of closed sets whose intersection avoids a compact set, there exists a finite subfamily whose intersection avoids this compact set.

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theorem locally_finite.finite_nonempty_inter_compact {α : Type u} [topological_space α] {ι : Type u_1} {f : ι → set α} (hf : locally_finite f) {s : set α} (hs : is_compact s) :
{i : ι | (f i ∩ s).nonempty}.finite

If s is a compact set in a topological space α and f : ι → set α is a locally finite family of sets, then f i ∩ s is nonempty only for a finitely many i.

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theorem is_compact.inter_Inter_nonempty {α : Type u} [topological_space α] {s : set α} {ι : Type v} (hs : is_compact s) (Z : ι → set α) (hZc : ∀ (i : ι), is_closed (Z i)) (hsZ : ∀ (t : finset ι), (s ∩ ⋂ (i : ι) (H : i ∈ t), Z i).nonempty) :
(s ∩ ⋂ (i : ι), Z i).nonempty

To show that a compact set intersects the intersection of a family of closed sets, it is sufficient to show that it intersects every finite subfamily.

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theorem is_compact.nonempty_Inter_of_directed_nonempty_compact_closed {α : Type u} [topological_space α] {ι : Type v} [hι : nonempty ι] (Z : ι → set α) (hZd : directed superset Z) (hZn : ∀ (i : ι), (Z i).nonempty) (hZc : ∀ (i : ι), is_compact (Z i)) (hZcl : ∀ (i : ι), is_closed (Z i)) :
(⋂ (i : ι), Z i).nonempty

Cantor's intersection theorem: the intersection of a directed family of nonempty compact closed sets is nonempty.

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theorem is_compact.nonempty_Inter_of_sequence_nonempty_compact_closed {α : Type u} [topological_space α] (Z : ℕ → set α) (hZd : ∀ (i : ℕ), Z (i + 1) ⊆ Z i) (hZn : ∀ (i : ℕ), (Z i).nonempty) (hZ0 : is_compact (Z 0)) (hZcl : ∀ (i : ℕ), is_closed (Z i)) :
(⋂ (i : ℕ), Z i).nonempty

Cantor's intersection theorem for sequences indexed by ℕ: the intersection of a decreasing sequence of nonempty compact closed sets is nonempty.

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theorem is_compact.elim_finite_subcover_image {α : Type u} {ι : Type u_1} [topological_space α] {s : set α} {b : set ι} {c : ι → set α} (hs : is_compact s) (hc₁ : ∀ (i : ι), i ∈ b → is_open (c i)) (hc₂ : s ⊆ ⋃ (i : ι) (H : i ∈ b), c i) :
∃ (b' : set ι) (H : b' ⊆ b), b'.finite ∧ s ⊆ ⋃ (i : ι) (H : i ∈ b'), c i

For every open cover of a compact set, there exists a finite subcover.

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theorem is_compact_of_finite_subfamily_closed {α : Type u} [topological_space α] {s : set α} (h : ∀ {ι : Type u} (Z : ι → set α), (∀ (i : ι), is_closed (Z i)) → (s ∩ ⋂ (i : ι), Z i) = ∅ → (∃ (t : finset ι), (s ∩ ⋂ (i : ι) (H : i ∈ t), Z i) = ∅)) :
is_compact s

A set s is compact if for every family of closed sets whose intersection avoids s, there exists a finite subfamily whose intersection avoids s.

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theorem is_compact_of_finite_subcover {α : Type u} [topological_space α] {s : set α} (h : ∀ {ι : Type u} (U : ι → set α), (∀ (i : ι), is_open (U i)) → (s ⊆ ⋃ (i : ι), U i) → (∃ (t : finset ι), s ⊆ ⋃ (i : ι) (H : i ∈ t), U i)) :
is_compact s

A set s is compact if for every open cover of s, there exists a finite subcover.

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theorem is_compact_iff_finite_subcover {α : Type u} [topological_space α] {s : set α} :
is_compact s ↔ ∀ {ι : Type u} (U : ι → set α), (∀ (i : ι), is_open (U i)) → (s ⊆ ⋃ (i : ι), U i) → (∃ (t : finset ι), s ⊆ ⋃ (i : ι) (H : i ∈ t), U i)

A set s is compact if and only if for every open cover of s, there exists a finite subcover.

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theorem is_compact_iff_finite_subfamily_closed {α : Type u} [topological_space α] {s : set α} :
is_compact s ↔ ∀ {ι : Type u} (Z : ι → set α), (∀ (i : ι), is_closed (Z i)) → (s ∩ ⋂ (i : ι), Z i) = ∅ → (∃ (t : finset ι), (s ∩ ⋂ (i : ι) (H : i ∈ t), Z i) = ∅)

A set s is compact if and only if for every family of closed sets whose intersection avoids s, there exists a finite subfamily whose intersection avoids s.

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theorem is_compact.eventually_forall_of_forall_eventually {α : Type u} {β : Type v} [topological_space α] [topological_space β] {x₀ : α} {K : set β} (hK : is_compact K) {P : α → β → Prop} (hP : ∀ (y : β), y ∈ K → (∀ᶠ (z : α × β) in 𝓝 (x₀, y), P z.fst z.snd)) :
∀ᶠ (x : α) in 𝓝 x₀, ∀ (y : β), y ∈ K → P x y

To show that ∀ y ∈ K, P x y holds for x close enough to x₀ when K is compact, it is sufficient to show that for all y₀ ∈ K there P x y holds for (x, y) close enough to (x₀, y₀).

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@[simp]
theorem is_compact_empty {α : Type u} [topological_space α] :
is_compact ∅
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@[simp]
theorem is_compact_singleton {α : Type u} [topological_space α] {a : α} :
is_compact {a}
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theorem set.subsingleton.is_compact {α : Type u} [topological_space α] {s : set α} (hs : s.subsingleton) :
is_compact s
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theorem set.finite.compact_bUnion {α : Type u} {ι : Type u_1} [topological_space α] {s : set ι} {f : ι → set α} (hs : s.finite) (hf : ∀ (i : ι), i ∈ s → is_compact (f i)) :
is_compact (⋃ (i : ι) (H : i ∈ s), f i)
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theorem finset.compact_bUnion {α : Type u} {ι : Type u_1} [topological_space α] (s : finset ι) {f : ι → set α} (hf : ∀ (i : ι), i ∈ s → is_compact (f i)) :
is_compact (⋃ (i : ι) (H : i ∈ s), f i)
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theorem compact_accumulate {α : Type u} [topological_space α] {K : ℕ → set α} (hK : ∀ (n : ℕ), is_compact (K n)) (n : ℕ) :
is_compact (set.accumulate K n)
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theorem compact_Union {α : Type u} {ι : Type u_1} [topological_space α] {f : ι → set α} [fintype ι] (h : ∀ (i : ι), is_compact (f i)) :
is_compact (⋃ (i : ι), f i)
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theorem set.finite.is_compact {α : Type u} [topological_space α] {s : set α} (hs : s.finite) :
is_compact s
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theorem finite_of_is_compact_of_discrete {α : Type u} [topological_space α] [discrete_topology α] (s : set α) (hs : is_compact s) :
s.finite
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theorem is_compact.union {α : Type u} [topological_space α] {s t : set α} (hs : is_compact s) (ht : is_compact t) :
is_compact (s ∪ t)
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theorem is_compact.insert {α : Type u} [topological_space α] {s : set α} (hs : is_compact s) (a : α) :
is_compact (insert a s)
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theorem exists_subset_nhd_of_compact' {α : Type u} [topological_space α] {ι : Type u_1} [nonempty ι] {V : ι → set α} (hV : directed superset V) (hV_cpct : ∀ (i : ι), is_compact (V i)) (hV_closed : ∀ (i : ι), is_closed (V i)) {U : set α} (hU : ∀ (x : α), (x ∈ ⋂ (i : ι), V i) → U ∈ 𝓝 x) :
∃ (i : ι), V i ⊆ U

If V : ι → set α is a decreasing family of closed compact sets then any neighborhood of ⋂ i, V i contains some V i. We assume each V i is compact and closed because α is not assumed to be Hausdorff. See exists_subset_nhd_of_compact for version assuming this.

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def filter.cocompact (α : Type u_1) [topological_space α] :
filter α

filter.cocompact is the filter generated by complements to compact sets.

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theorem filter.has_basis_cocompact {α : Type u} [topological_space α] :
(filter.cocompact α).has_basis is_compact compl
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theorem filter.mem_cocompact {α : Type u} [topological_space α] {s : set α} :
s ∈ filter.cocompact α ↔ ∃ (t : set α), is_compact t ∧ tᶜ ⊆ s
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theorem filter.mem_cocompact' {α : Type u} [topological_space α] {s : set α} :
s ∈ filter.cocompact α ↔ ∃ (t : set α), is_compact t ∧ sᶜ ⊆ t
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theorem is_compact.compl_mem_cocompact {α : Type u} [topological_space α] {s : set α} (hs : is_compact s) :
sᶜ ∈ filter.cocompact α
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def filter.coclosed_compact (α : Type u_1) [topological_space α] :
filter α

filter.coclosed_compact is the filter generated by complements to closed compact sets. In a Hausdorff space, this is the same as filter.cocompact.

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theorem filter.has_basis_coclosed_compact {α : Type u} [topological_space α] :
(filter.coclosed_compact α).has_basis (λ (s : set α), is_closed s ∧ is_compact s) compl
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theorem filter.mem_coclosed_compact {α : Type u} [topological_space α] {s : set α} :
s ∈ filter.coclosed_compact α ↔ ∃ (t : set α), is_closed t ∧ is_compact t ∧ tᶜ ⊆ s
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theorem filter.mem_coclosed_compact' {α : Type u} [topological_space α] {s : set α} :
s ∈ filter.coclosed_compact α ↔ ∃ (t : set α), is_closed t ∧ is_compact t ∧ sᶜ ⊆ t
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theorem filter.cocompact_le_coclosed_compact {α : Type u} [topological_space α] :
filter.cocompact α ≤ filter.coclosed_compact α
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def nhds_contain_boxes {α : Type u} {β : Type v} [topological_space α] [topological_space β] (s : set α) (t : set β) :
Prop

nhds_contain_boxes s t means that any open neighborhood of s × t in α × β includes a product of an open neighborhood of s by an open neighborhood of t.

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theorem nhds_contain_boxes.symm {α : Type u} {β : Type v} [topological_space α] [topological_space β] {s : set α} {t : set β} :
nhds_contain_boxes s t → nhds_contain_boxes t s
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theorem nhds_contain_boxes.comm {α : Type u} {β : Type v} [topological_space α] [topological_space β] {s : set α} {t : set β} :
nhds_contain_boxes s t ↔ nhds_contain_boxes t s
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theorem nhds_contain_boxes_of_singleton {α : Type u} {β : Type v} [topological_space α] [topological_space β] {x : α} {y : β} :
nhds_contain_boxes {x} {y}
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theorem nhds_contain_boxes_of_compact {α : Type u} {β : Type v} [topological_space α] [topological_space β] {s : set α} (hs : is_compact s) (t : set β) (H : ∀ (x : α), x ∈ s → nhds_contain_boxes {x} t) :
nhds_contain_boxes s t
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theorem generalized_tube_lemma {α : Type u} {β : Type v} [topological_space α] [topological_space β] {s : set α} (hs : is_compact s) {t : set β} (ht : is_compact t) {n : set (α × β)} (hn : is_open n) (hp : s ×ˢ t ⊆ n) :
∃ (u : set α) (v : set β), is_open u ∧ is_open v ∧ s ⊆ u ∧ t ⊆ v ∧ u ×ˢ v ⊆ n

If s and t are compact sets and n is an open neighborhood of s × t, then there exist open neighborhoods u ⊇ s and v ⊇ t such that u × v ⊆ n.

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@[class]
structure compact_space (α : Type u_3) [topological_space α] :
Prop

Type class for compact spaces. Separation is sometimes included in the definition, especially in the French literature, but we do not include it here.

Instances
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@[protected, instance]
def subsingleton.compact_space {α : Type u} [topological_space α] [subsingleton α] :
compact_space α
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theorem is_compact_univ_iff {α : Type u} [topological_space α] :
is_compact set.univ ↔ compact_space α
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theorem compact_univ {α : Type u} [topological_space α] [h : compact_space α] :
is_compact set.univ
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theorem cluster_point_of_compact {α : Type u} [topological_space α] [compact_space α] (f : filter α) [f.ne_bot] :
∃ (x : α), cluster_pt x f
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theorem compact_space.elim_nhds_subcover {α : Type u} [topological_space α] [compact_space α] (U : α → set α) (hU : ∀ (x : α), U x ∈ 𝓝 x) :
∃ (t : finset α), (⋃ (x : α) (H : x ∈ t), U x) = ⊤
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theorem compact_space_of_finite_subfamily_closed {α : Type u} [topological_space α] (h : ∀ {ι : Type u} (Z : ι → set α), (∀ (i : ι), is_closed (Z i)) → (⋂ (i : ι), Z i) = ∅ → (∃ (t : finset ι), (⋂ (i : ι) (H : i ∈ t), Z i) = ∅)) :
compact_space α
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theorem is_closed.is_compact {α : Type u} [topological_space α] [compact_space α] {s : set α} (h : is_closed s) :
is_compact s
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@[class]
structure noncompact_space (α : Type u_3) [topological_space α] :
Prop

α is a noncompact topological space if it not a compact space.

Instances
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theorem is_compact.ne_univ {α : Type u} [topological_space α] [noncompact_space α] {s : set α} (hs : is_compact s) :
s ≠ set.univ
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@[protected, instance]
def filter.cocompact.filter.ne_bot {α : Type u} [topological_space α] [noncompact_space α] :
(filter.cocompact α).ne_bot
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@[protected, instance]
def filter.coclosed_compact.filter.ne_bot {α : Type u} [topological_space α] [noncompact_space α] :
(filter.coclosed_compact α).ne_bot
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theorem noncompact_space_of_ne_bot {α : Type u} [topological_space α] (h : (filter.cocompact α).ne_bot) :
noncompact_space α
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theorem filter.cocompact_ne_bot_iff {α : Type u} [topological_space α] :
(filter.cocompact α).ne_bot ↔ noncompact_space α
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theorem not_compact_space_iff {α : Type u} [topological_space α] :
¬compact_space α ↔ noncompact_space α
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noncomputable def fintype_of_compact_of_discrete {α : Type u} [topological_space α] [compact_space α] [discrete_topology α] :
fintype α

A compact discrete space is finite.

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theorem finite_cover_nhds_interior {α : Type u} [topological_space α] [compact_space α] {U : α → set α} (hU : ∀ (x : α), U x ∈ 𝓝 x) :
∃ (t : finset α), (⋃ (x : α) (H : x ∈ t), interior (U x)) = set.univ
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theorem finite_cover_nhds {α : Type u} [topological_space α] [compact_space α] {U : α → set α} (hU : ∀ (x : α), U x ∈ 𝓝 x) :
∃ (t : finset α), (⋃ (x : α) (H : x ∈ t), U x) = set.univ
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theorem locally_finite.finite_nonempty_of_compact {α : Type u} [topological_space α] {ι : Type u_1} [compact_space α] {f : ι → set α} (hf : locally_finite f) :
{i : ι | (f i).nonempty}.finite

If α is a compact space, then a locally finite family of sets of α can have only finitely many nonempty elements.

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theorem locally_finite.finite_of_compact {α : Type u} [topological_space α] {ι : Type u_1} [compact_space α] {f : ι → set α} (hf : locally_finite f) (hne : ∀ (i : ι), (f i).nonempty) :
set.univ.finite

If α is a compact space, then a locally finite family of nonempty sets of α can have only finitely many elements, set.finite version.

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noncomputable def locally_finite.fintype_of_compact {α : Type u} [topological_space α] {ι : Type u_1} [compact_space α] {f : ι → set α} (hf : locally_finite f) (hne : ∀ (i : ι), (f i).nonempty) :
fintype ι

If α is a compact space, then a locally finite family of nonempty sets of α can have only finitely many elements, fintype version.

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theorem is_compact.image_of_continuous_on {α : Type u} {β : Type v} [topological_space α] [topological_space β] {s : set α} {f : α → β} (hs : is_compact s) (hf : continuous_on f s) :
is_compact (f '' s)
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theorem is_compact.image {α : Type u} {β : Type v} [topological_space α] [topological_space β] {s : set α} {f : α → β} (hs : is_compact s) (hf : continuous f) :
is_compact (f '' s)
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theorem filter.comap_cocompact {α : Type u} {β : Type v} [topological_space α] [topological_space β] {f : α → β} (hf : continuous f) :
filter.comap f (filter.cocompact β) ≤ filter.cocompact α

The comap of the cocompact filter on β by a continuous function f : α → β is less than or equal to the cocompact filter on α. This is a reformulation of the fact that images of compact sets are compact.

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theorem is_compact_range {α : Type u} {β : Type v} [topological_space α] [topological_space β] [compact_space α] {f : α → β} (hf : continuous f) :
is_compact (set.range f)
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theorem is_closed_proj_of_is_compact {X : Type u_1} [topological_space X] [compact_space X] {Y : Type u_2} [topological_space Y] :
is_closed_map prod.snd

If X is is_compact then pr₂ : X × Y → Y is a closed map

source
theorem exists_subset_nhd_of_compact_space {α : Type u} [topological_space α] [compact_space α] {ι : Type u_1} [nonempty ι] {V : ι → set α} (hV : directed superset V) (hV_closed : ∀ (i : ι), is_closed (V i)) {U : set α} (hU : ∀ (x : α), (x ∈ ⋂ (i : ι), V i) → U ∈ 𝓝 x) :
∃ (i : ι), V i ⊆ U
source
theorem inducing.is_compact_iff {α : Type u} {β : Type v} [topological_space α] [topological_space β] {f : α → β} (hf : inducing f) {s : set α} :
is_compact (f '' s) ↔ is_compact s

If f : α → β is an inducing map, then the image f '' s of a set s is compact if and only if the set s is closed.

source
theorem embedding.is_compact_iff_is_compact_image {α : Type u} {β : Type v} [topological_space α] [topological_space β] {s : set α} {f : α → β} (hf : embedding f) :
is_compact s ↔ is_compact (f '' s)

If f : α → β is an embedding (or more generally, an inducing map, see inducing.is_compact_iff), then the image f '' s of a set s is compact if and only if the set s is closed.

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theorem closed_embedding.is_compact_preimage {α : Type u} {β : Type v} [topological_space α] [topological_space β] {f : α → β} (hf : closed_embedding f) {K : set β} (hK : is_compact K) :
is_compact (f ⁻¹' K)

The preimage of a compact set under a closed embedding is a compact set.

source
theorem closed_embedding.tendsto_cocompact {α : Type u} {β : Type v} [topological_space α] [topological_space β] {f : α → β} (hf : closed_embedding f) :
filter.tendsto f (filter.cocompact α) (filter.cocompact β)

A closed embedding is proper, ie, inverse images of compact sets are contained in compacts. Moreover, the preimage of a compact set is compact, see closed_embedding.is_compact_preimage.

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theorem compact_iff_compact_in_subtype {α : Type u} [topological_space α] {p : α → Prop} {s : set {a // p a}} :
is_compact s ↔ is_compact (coe '' s)
source
theorem is_compact_iff_is_compact_univ {α : Type u} [topological_space α] {s : set α} :
is_compact s ↔ is_compact set.univ
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theorem is_compact_iff_compact_space {α : Type u} [topological_space α] {s : set α} :
is_compact s ↔ compact_space ↥s
source
@[protected]
theorem closed_embedding.noncompact_space {α : Type u} {β : Type v} [topological_space α] [topological_space β] [noncompact_space α] {f : α → β} (hf : closed_embedding f) :
noncompact_space β
source
@[protected]
theorem closed_embedding.compact_space {α : Type u} {β : Type v} [topological_space α] [topological_space β] [h : compact_space β] {f : α → β} (hf : closed_embedding f) :
compact_space α
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theorem is_compact.prod {α : Type u} {β : Type v} [topological_space α] [topological_space β] {s : set α} {t : set β} (hs : is_compact s) (ht : is_compact t) :
is_compact (s ×ˢ t)
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@[protected, instance]
def fintype.compact_space {α : Type u} [topological_space α] [fintype α] :
compact_space α

Finite topological spaces are compact.

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@[protected, instance]
def prod.compact_space {α : Type u} {β : Type v} [topological_space α] [topological_space β] [compact_space α] [compact_space β] :
compact_space (α × β)

The product of two compact spaces is compact.

source
@[protected, instance]
def sum.compact_space {α : Type u} {β : Type v} [topological_space α] [topological_space β] [compact_space α] [compact_space β] :
compact_space (α ⊕ β)

The disjoint union of two compact spaces is compact.

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@[protected, instance]
def sigma.compact_space {ι : Type u_1} {π : ι → Type u_2} [fintype ι] [Π (i : ι), topological_space (π i)] [∀ (i : ι), compact_space (π i)] :
compact_space (Σ (i : ι), π i)
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theorem filter.coprod_cocompact {α : Type u} {β : Type v} [topological_space α] [topological_space β] :
(filter.cocompact α).coprod (filter.cocompact β) = filter.cocompact (α × β)

The coproduct of the cocompact filters on two topological spaces is the cocompact filter on their product.

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theorem prod.noncompact_space_iff {α : Type u} {β : Type v} [topological_space α] [topological_space β] :
noncompact_space (α × β) ↔ noncompact_space α ∧ nonempty β ∨ nonempty α ∧ noncompact_space β
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@[protected, instance]
def prod.noncompact_space_left {α : Type u} {β : Type v} [topological_space α] [topological_space β] [noncompact_space α] [nonempty β] :
noncompact_space (α × β)
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@[protected, instance]
def prod.noncompact_space_right {α : Type u} {β : Type v} [topological_space α] [topological_space β] [nonempty α] [noncompact_space β] :
noncompact_space (α × β)
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theorem is_compact_pi_infinite {ι : Type u_1} {π : ι → Type u_2} [Π (i : ι), topological_space (π i)] {s : Π (i : ι), set (π i)} :
(∀ (i : ι), is_compact (s i)) → is_compact {x : Π (i : ι), π i | ∀ (i : ι), x i ∈ s i}

Tychonoff's theorem

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theorem is_compact_univ_pi {ι : Type u_1} {π : ι → Type u_2} [Π (i : ι), topological_space (π i)] {s : Π (i : ι), set (π i)} (h : ∀ (i : ι), is_compact (s i)) :
is_compact (set.univ.pi s)

A version of Tychonoff's theorem that uses set.pi.

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@[protected, instance]
def pi.compact_space {ι : Type u_1} {π : ι → Type u_2} [Π (i : ι), topological_space (π i)] [∀ (i : ι), compact_space (π i)] :
compact_space (Π (i : ι), π i)
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theorem filter.Coprod_cocompact {δ : Type u_1} {κ : δ → Type u_2} [Π (d : δ), topological_space (κ d)] :
filter.Coprod (λ (d : δ), filter.cocompact (κ d)) = filter.cocompact (Π (d : δ), κ d)

Product of compact sets is compact

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@[protected, instance]
def quot.compact_space {α : Type u} [topological_space α] {r : α → α → Prop} [compact_space α] :
compact_space (quot r)
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@[protected, instance]
def quotient.compact_space {α : Type u} [topological_space α] {s : setoid α} [compact_space α] :
compact_space (quotient s)
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@[class]
structure locally_compact_space (α : Type u_3) [topological_space α] :
Prop

There are various definitions of "locally compact space" in the literature, which agree for Hausdorff spaces but not in general. This one is the precise condition on X needed for the evaluation map C(X, Y) × X → Y to be continuous for all Y when C(X, Y) is given the compact-open topology.

Instances
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theorem compact_basis_nhds {α : Type u} [topological_space α] [locally_compact_space α] (x : α) :
(𝓝 x).has_basis (λ (s : set α), s ∈ 𝓝 x ∧ is_compact s) (λ (s : set α), s)
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theorem locally_compact_space_of_has_basis {α : Type u} [topological_space α] {ι : α → Type u_1} {p : Π (x : α), ι x → Prop} {s : Π (x : α), ι x → set α} (h : ∀ (x : α), (𝓝 x).has_basis (p x) (s x)) (hc : ∀ (x : α) (i : ι x), p x i → is_compact (s x i)) :
locally_compact_space α
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@[protected, instance]
def locally_compact_space.prod (α : Type u_1) (β : Type u_2) [topological_space α] [topological_space β] [locally_compact_space α] [locally_compact_space β] :
locally_compact_space (α × β)
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theorem exists_compact_subset {α : Type u} [topological_space α] [locally_compact_space α] {x : α} {U : set α} (hU : is_open U) (hx : x ∈ U) :
∃ (K : set α), is_compact K ∧ x ∈ interior K ∧ K ⊆ U

A reformulation of the definition of locally compact space: In a locally compact space, every open set containing x has a compact subset containing x in its interior.

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theorem exists_compact_mem_nhds {α : Type u} [topological_space α] [locally_compact_space α] (x : α) :
∃ (K : set α), is_compact K ∧ K ∈ 𝓝 x

In a locally compact space every point has a compact neighborhood.

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theorem exists_compact_superset {α : Type u} [topological_space α] [locally_compact_space α] {K : set α} (hK : is_compact K) :
∃ (K' : set α), is_compact K' ∧ K ⊆ interior K'

In a locally compact space, every compact set is contained in the interior of a compact set.

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@[protected]
theorem closed_embedding.locally_compact_space {α : Type u} {β : Type v} [topological_space α] [topological_space β] [locally_compact_space β] {f : α → β} (hf : closed_embedding f) :
locally_compact_space α
source
@[protected]
theorem is_closed.locally_compact_space {α : Type u} [topological_space α] [locally_compact_space α] {s : set α} (hs : is_closed s) :
locally_compact_space ↥s
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@[protected]
theorem open_embedding.locally_compact_space {α : Type u} {β : Type v} [topological_space α] [topological_space β] [locally_compact_space β] {f : α → β} (hf : open_embedding f) :
locally_compact_space α
source
@[protected]
theorem is_open.locally_compact_space {α : Type u} [topological_space α] [locally_compact_space α] {s : set α} (hs : is_open s) :
locally_compact_space ↥s
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theorem ultrafilter.le_nhds_Lim {α : Type u} [topological_space α] [compact_space α] (F : ultrafilter α) :
↑F ≤ 𝓝 (Lim ↑F)
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theorem is_closed.exists_minimal_nonempty_closed_subset {α : Type u} [topological_space α] [compact_space α] {S : set α} (hS : is_closed S) (hne : S.nonempty) :
∃ (V : set α), V ⊆ S ∧ V.nonempty ∧ is_closed V ∧ ∀ (V' : set α), V' ⊆ V → V'.nonempty → is_closed V' → V' = V
source
@[class]
structure sigma_compact_space (α : Type u_3) [topological_space α] :
Prop

A σ-compact space is a space that is the union of a countable collection of compact subspaces. Note that a locally compact separable T₂ space need not be σ-compact. The sequence can be extracted using topological_space.compact_covering.

Instances
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@[protected, instance]
def compact_space.sigma_compact {α : Type u} [topological_space α] [compact_space α] :
sigma_compact_space α
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theorem sigma_compact_space.of_countable {α : Type u} [topological_space α] (S : set (set α)) (Hc : S.countable) (Hcomp : ∀ (s : set α), s ∈ S → is_compact s) (HU : ⋃₀S = set.univ) :
sigma_compact_space α
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@[protected, instance]
def sigma_compact_space_of_locally_compact_second_countable {α : Type u} [topological_space α] [locally_compact_space α] [topological_space.second_countable_topology α] :
sigma_compact_space α
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def compact_covering (α : Type u) [topological_space α] [sigma_compact_space α] :
ℕ → set α

A choice of compact covering for a σ-compact space, chosen to be monotone.

Equations
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theorem is_compact_compact_covering (α : Type u) [topological_space α] [sigma_compact_space α] (n : ℕ) :
is_compact (compact_covering α n)
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theorem Union_compact_covering (α : Type u) [topological_space α] [sigma_compact_space α] :
(⋃ (n : ℕ), compact_covering α n) = set.univ
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theorem compact_covering_subset (α : Type u) [topological_space α] [sigma_compact_space α] ⦃m n : ℕ⦄ (h : m ≤ n) :
compact_covering α m ⊆ compact_covering α n
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theorem exists_mem_compact_covering {α : Type u} [topological_space α] [sigma_compact_space α] (x : α) :
∃ (n : ℕ), x ∈ compact_covering α n
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@[protected]
theorem locally_finite.countable_univ {α : Type u} [topological_space α] [sigma_compact_space α] {ι : Type u_1} {f : ι → set α} (hf : locally_finite f) (hne : ∀ (i : ι), (f i).nonempty) :
set.univ.countable

If α is a σ-compact space, then a locally finite family of nonempty sets of α can have only countably many elements, set.countable version.

source
@[protected]
noncomputable def locally_finite.encodable {α : Type u} [topological_space α] [sigma_compact_space α] {ι : Type u_1} {f : ι → set α} (hf : locally_finite f) (hne : ∀ (i : ι), (f i).nonempty) :
encodable ι

If f : ι → set α is a locally finite covering of a σ-compact topological space by nonempty sets, then the index type ι is encodable.

Equations
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theorem countable_cover_nhds_within_of_sigma_compact {α : Type u} [topological_space α] [sigma_compact_space α] {f : α → set α} {s : set α} (hs : is_closed s) (hf : ∀ (x : α), x ∈ s → f x ∈ 𝓝[s] x) :
∃ (t : set α) (H : t ⊆ s), t.countable ∧ s ⊆ ⋃ (x : α) (H : x ∈ t), f x

In a topological space with sigma compact topology, if f is a function that sends each point x of a closed set s to a neighborhood of x within s, then for some countable set t ⊆ s, the neighborhoods f x, x ∈ t, cover the whole set s.

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theorem countable_cover_nhds_of_sigma_compact {α : Type u} [topological_space α] [sigma_compact_space α] {f : α → set α} (hf : ∀ (x : α), f x ∈ 𝓝 x) :
∃ (s : set α), s.countable ∧ (⋃ (x : α) (H : x ∈ s), f x) = set.univ

In a topological space with sigma compact topology, if f is a function that sends each point x to a neighborhood of x, then for some countable set s, the neighborhoods f x, x ∈ s, cover the whole space.

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structure compact_exhaustion (X : Type u_3) [topological_space X] :
Type u_3

An exhaustion by compact sets of a topological space is a sequence of compact sets K n such that K n ⊆ interior (K (n + 1)) and (⋃ n, K n) = univ.

If X is a locally compact sigma compact space, then compact_exhaustion.choice X provides a choice of an exhaustion by compact sets. This choice is also available as (default : compact_exhaustion X).

source
@[protected, instance]
def compact_exhaustion.has_coe_to_fun {α : Type u} [topological_space α] :
has_coe_to_fun (compact_exhaustion α) (λ (_x : compact_exhaustion α), ℕ → set α)
Equations
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@[protected]
theorem compact_exhaustion.is_compact {α : Type u} [topological_space α] (K : compact_exhaustion α) (n : ℕ) :
is_compact (⇑K n)
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theorem compact_exhaustion.subset_interior_succ {α : Type u} [topological_space α] (K : compact_exhaustion α) (n : ℕ) :
⇑K n ⊆ interior (⇑K (n + 1))
source
theorem compact_exhaustion.subset_succ {α : Type u} [topological_space α] (K : compact_exhaustion α) (n : ℕ) :
⇑K n ⊆ ⇑K (n + 1)
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@[protected]
theorem compact_exhaustion.subset {α : Type u} [topological_space α] (K : compact_exhaustion α) ⦃m n : ℕ⦄ (h : m ≤ n) :
⇑K m ⊆ ⇑K n
source
theorem compact_exhaustion.subset_interior {α : Type u} [topological_space α] (K : compact_exhaustion α) ⦃m n : ℕ⦄ (h : m < n) :
⇑K m ⊆ interior (⇑K n)
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theorem compact_exhaustion.Union_eq {α : Type u} [topological_space α] (K : compact_exhaustion α) :
(⋃ (n : ℕ), ⇑K n) = set.univ
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theorem compact_exhaustion.exists_mem {α : Type u} [topological_space α] (K : compact_exhaustion α) (x : α) :
∃ (n : ℕ), x ∈ ⇑K n
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@[protected]
noncomputable def compact_exhaustion.find {α : Type u} [topological_space α] (K : compact_exhaustion α) (x : α) :
ℕ

The minimal n such that x ∈ K n.

Equations
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theorem compact_exhaustion.mem_find {α : Type u} [topological_space α] (K : compact_exhaustion α) (x : α) :
x ∈ ⇑K (K.find x)
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theorem compact_exhaustion.mem_iff_find_le {α : Type u} [topological_space α] (K : compact_exhaustion α) {x : α} {n : ℕ} :
x ∈ ⇑K n ↔ K.find x ≤ n
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def compact_exhaustion.shiftr {α : Type u} [topological_space α] (K : compact_exhaustion α) :
compact_exhaustion α

Prepend the empty set to a compact exhaustion K n.

Equations
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@[simp]
theorem compact_exhaustion.find_shiftr {α : Type u} [topological_space α] (K : compact_exhaustion α) (x : α) :
K.shiftr.find x = K.find x + 1
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theorem compact_exhaustion.mem_diff_shiftr_find {α : Type u} [topological_space α] (K : compact_exhaustion α) (x : α) :
x ∈ ⇑(K.shiftr) (K.find x + 1) \ ⇑(K.shiftr) (K.find x)
source
noncomputable def compact_exhaustion.choice (X : Type u_1) [topological_space X] [locally_compact_space X] [sigma_compact_space X] :
compact_exhaustion X

A choice of an exhaustion by compact sets of a locally compact sigma compact space.

Equations
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@[protected, instance]
noncomputable def compact_exhaustion.inhabited {α : Type u} [topological_space α] [locally_compact_space α] [sigma_compact_space α] :
inhabited (compact_exhaustion α)
Equations
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def is_clopen {α : Type u} [topological_space α] (s : set α) :
Prop

A set is clopen if it is both open and closed.

Equations
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@[protected]
theorem is_clopen.is_open {α : Type u} [topological_space α] {s : set α} (hs : is_clopen s) :
is_open s
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@[protected]
theorem is_clopen.is_closed {α : Type u} [topological_space α] {s : set α} (hs : is_clopen s) :
is_closed s
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theorem is_clopen.union {α : Type u} [topological_space α] {s t : set α} (hs : is_clopen s) (ht : is_clopen t) :
is_clopen (s ∪ t)
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theorem is_clopen.inter {α : Type u} [topological_space α] {s t : set α} (hs : is_clopen s) (ht : is_clopen t) :
is_clopen (s ∩ t)
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@[simp]
theorem is_clopen_empty {α : Type u} [topological_space α] :
is_clopen ∅
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@[simp]
theorem is_clopen_univ {α : Type u} [topological_space α] :
is_clopen set.univ
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theorem is_clopen.compl {α : Type u} [topological_space α] {s : set α} (hs : is_clopen s) :
is_clopen sᶜ
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@[simp]
theorem is_clopen_compl_iff {α : Type u} [topological_space α] {s : set α} :
is_clopen sᶜ ↔ is_clopen s
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theorem is_clopen.diff {α : Type u} [topological_space α] {s t : set α} (hs : is_clopen s) (ht : is_clopen t) :
is_clopen (s \ t)
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theorem is_clopen_Union {α : Type u} [topological_space α] {β : Type u_1} [fintype β] {s : β → set α} (h : ∀ (i : β), is_clopen (s i)) :
is_clopen (⋃ (i : β), s i)
source
theorem is_clopen_bUnion {α : Type u} [topological_space α] {β : Type u_1} {s : finset β} {f : β → set α} (h : ∀ (i : β), i ∈ s → is_clopen (f i)) :
is_clopen (⋃ (i : β) (H : i ∈ s), f i)
source
theorem is_clopen_Inter {α : Type u} [topological_space α] {β : Type u_1} [fintype β] {s : β → set α} (h : ∀ (i : β), is_clopen (s i)) :
is_clopen (⋂ (i : β), s i)
source
theorem is_clopen_bInter {α : Type u} [topological_space α] {β : Type u_1} {s : finset β} {f : β → set α} (h : ∀ (i : β), i ∈ s → is_clopen (f i)) :
is_clopen (⋂ (i : β) (H : i ∈ s), f i)
source
theorem continuous_on.preimage_clopen_of_clopen {α : Type u} {β : Type v} [topological_space α] [topological_space β] {f : α → β} {s : set α} {t : set β} (hf : continuous_on f s) (hs : is_clopen s) (ht : is_clopen t) :
is_clopen (s ∩ f ⁻¹' t)
source
theorem is_clopen_inter_of_disjoint_cover_clopen {α : Type u} [topological_space α] {Z a b : set α} (h : is_clopen Z) (cover : Z ⊆ a ∪ b) (ha : is_open a) (hb : is_open b) (hab : a ∩ b = ∅) :
is_clopen (Z ∩ a)

The intersection of a disjoint covering by two open sets of a clopen set will be clopen.

source
@[simp]
theorem is_clopen_discrete {α : Type u} [topological_space α] [discrete_topology α] (x : set α) :
is_clopen x
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theorem clopen_range_sigma_mk {ι : Type u_1} {σ : ι → Type u_2} [Π (i : ι), topological_space (σ i)] {i : ι} :
is_clopen (set.range (sigma.mk i))
source
@[protected]
theorem quotient_map.is_clopen_preimage {α : Type u} {β : Type v} [topological_space α] [topological_space β] {f : α → β} (hf : quotient_map f) {s : set β} :
is_clopen (f ⁻¹' s) ↔ is_clopen s
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def is_preirreducible {α : Type u} [topological_space α] (s : set α) :
Prop

A preirreducible set s is one where there is no non-trivial pair of disjoint opens on s.

Equations
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def is_irreducible {α : Type u} [topological_space α] (s : set α) :
Prop

An irreducible set s is one that is nonempty and where there is no non-trivial pair of disjoint opens on s.

Equations
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theorem is_irreducible.nonempty {α : Type u} [topological_space α] {s : set α} (h : is_irreducible s) :
s.nonempty
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theorem is_irreducible.is_preirreducible {α : Type u} [topological_space α] {s : set α} (h : is_irreducible s) :
is_preirreducible s
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theorem is_preirreducible_empty {α : Type u} [topological_space α] :
is_preirreducible ∅
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theorem is_irreducible_singleton {α : Type u} [topological_space α] {x : α} :
is_irreducible {x}
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theorem is_preirreducible.closure {α : Type u} [topological_space α] {s : set α} (H : is_preirreducible s) :
is_preirreducible (closure s)
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theorem is_irreducible.closure {α : Type u} [topological_space α] {s : set α} (h : is_irreducible s) :
is_irreducible (closure s)
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theorem is_preirreducible_of_subsingleton {α : Type u} [topological_space α] (s : set α) [hs : subsingleton ↥s] :
is_preirreducible s
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theorem exists_preirreducible {α : Type u} [topological_space α] (s : set α) (H : is_preirreducible s) :
∃ (t : set α), is_preirreducible t ∧ s ⊆ t ∧ ∀ (u : set α), is_preirreducible u → t ⊆ u → u = t
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def irreducible_component {α : Type u} [topological_space α] (x : α) :
set α

A maximal irreducible set that contains a given point.

Equations
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theorem irreducible_component_property {α : Type u} [topological_space α] (x : α) :
is_preirreducible (irreducible_component x) ∧ {x} ⊆ irreducible_component x ∧ ∀ (u : set α), is_preirreducible u → irreducible_component x ⊆ u → u = irreducible_component x
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theorem mem_irreducible_component {α : Type u} [topological_space α] {x : α} :
x ∈ irreducible_component x
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theorem is_irreducible_irreducible_component {α : Type u} [topological_space α] {x : α} :
is_irreducible (irreducible_component x)
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theorem eq_irreducible_component {α : Type u} [topological_space α] {x : α} {s : set α} :
is_preirreducible s → irreducible_component x ⊆ s → s = irreducible_component x
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theorem is_closed_irreducible_component {α : Type u} [topological_space α] {x : α} :
is_closed (irreducible_component x)
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@[class]
structure preirreducible_space (α : Type u) [topological_space α] :
Prop

A preirreducible space is one where there is no non-trivial pair of disjoint opens.

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@[class]
structure irreducible_space (α : Type u) [topological_space α] :
Prop

An irreducible space is one that is nonempty and where there is no non-trivial pair of disjoint opens.

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theorem irreducible_space.is_irreducible_univ (α : Type u) [topological_space α] [irreducible_space α] :
is_irreducible ⊤
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theorem irreducible_space_def (α : Type u) [topological_space α] :
irreducible_space α ↔ is_irreducible ⊤
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theorem nonempty_preirreducible_inter {α : Type u} [topological_space α] [preirreducible_space α] {s t : set α} :
is_open s → is_open t → s.nonempty → t.nonempty → (s ∩ t).nonempty
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theorem is_preirreducible.image {α : Type u} {β : Type v} [topological_space α] [topological_space β] {s : set α} (H : is_preirreducible s) (f : α → β) (hf : continuous_on f s) :
is_preirreducible (f '' s)
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theorem is_irreducible.image {α : Type u} {β : Type v} [topological_space α] [topological_space β] {s : set α} (H : is_irreducible s) (f : α → β) (hf : continuous_on f s) :
is_irreducible (f '' s)
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theorem subtype.preirreducible_space {α : Type u} [topological_space α] {s : set α} (h : is_preirreducible s) :
preirreducible_space ↥s
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theorem subtype.irreducible_space {α : Type u} [topological_space α] {s : set α} (h : is_irreducible s) :
irreducible_space ↥s
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theorem is_irreducible_iff_sInter {α : Type u} [topological_space α] {s : set α} :
is_irreducible s ↔ ∀ (U : finset (set α)), (∀ (u : set α), u ∈ U → is_open u) → (∀ (u : set α), u ∈ U → (s ∩ u).nonempty) → (s ∩ ⋂₀↑U).nonempty

A set s is irreducible if and only if for every finite collection of open sets all of whose members intersect s, s also intersects the intersection of the entire collection (i.e., there is an element of s contained in every member of the collection).

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theorem is_preirreducible_iff_closed_union_closed {α : Type u} [topological_space α] {s : set α} :
is_preirreducible s ↔ ∀ (z₁ z₂ : set α), is_closed z₁ → is_closed z₂ → s ⊆ z₁ ∪ z₂ → s ⊆ z₁ ∨ s ⊆ z₂

A set is preirreducible if and only if for every cover by two closed sets, it is contained in one of the two covering sets.

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theorem is_irreducible_iff_sUnion_closed {α : Type u} [topological_space α] {s : set α} :
is_irreducible s ↔ ∀ (Z : finset (set α)), (∀ (z : set α), z ∈ Z → is_closed z) → s ⊆ ⋃₀↑Z → (∃ (z : set α) (H : z ∈ Z), s ⊆ z)

A set is irreducible if and only if for every cover by a finite collection of closed sets, it is contained in one of the members of the collection.

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theorem subset_closure_inter_of_is_preirreducible_of_is_open {α : Type u} [topological_space α] {S U : set α} (hS : is_preirreducible S) (hU : is_open U) (h : (S ∩ U).nonempty) :
S ⊆ closure (S ∩ U)

A nonemtpy open subset of a preirreducible subspace is dense in the subspace.

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theorem is_preirreducible.subset_irreducible {α : Type u} [topological_space α] {S U Z : set α} (hZ : is_preirreducible Z) (hU : U.nonempty) (hU' : is_open U) (h₁ : U ⊆ S) (h₂ : S ⊆ Z) :
is_irreducible S

If ∅ ≠ U ⊆ S ⊆ Z such that U is open and Z is preirreducible, then S is irreducible.

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theorem is_preirreducible.open_subset {α : Type u} [topological_space α] {Z U : set α} (hZ : is_preirreducible Z) (hU : is_open U) (hU' : U ⊆ Z) :
is_preirreducible U
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theorem is_preirreducible.interior {α : Type u} [topological_space α] {Z : set α} (hZ : is_preirreducible Z) :
is_preirreducible (interior Z)
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theorem is_preirreducible.preimage {α : Type u} {β : Type v} [topological_space α] [topological_space β] {Z : set α} (hZ : is_preirreducible Z) {f : β → α} (hf : open_embedding f) :
is_preirreducible (f ⁻¹' Z)