The OnePoint Compactification #
We construct the OnePoint compactification (the one-point compactification) of an arbitrary
topological space X
and prove some properties inherited from X
.
Main definitions #
OnePoint
: the OnePoint compactification, we use coercion for the canonical embeddingX → OnePoint X
; whenX
is already compact, the compactification adds an isolated point to the space.OnePoint.infty
: the extra point
Main results #
- The topological structure of
OnePoint X
- The connectedness of
OnePoint X
for a noncompact, preconnectedX
OnePoint X
isT₀
for a T₀ spaceX
OnePoint X
isT₁
for a T₁ spaceX
OnePoint X
is normal ifX
is a locally compact Hausdorff space
Tags #
one-point compactification, compactness
Definition and basic properties #
In this section we define OnePoint X
to be the disjoint union of X
and ∞
, implemented as
Option X
. Then we restate some lemmas about Option X
for OnePoint X
.
The point at infinity
Instances For
Topological space structure on OnePoint X
#
We define a topological space structure on OnePoint X
so that s
is open if and only if
(↑) ⁻¹' s
is open inX
;- if
∞ ∈ s
, then((↑) ⁻¹' s)ᶜ
is compact.
Then we reformulate this definition in a few different ways, and prove that
(↑) : X → OnePoint X
is an open embedding. If X
is not a compact space, then we also prove
that (↑)
has dense range, so it is a dense embedding.
An open set in OnePoint X
constructed from a closed compact set in X
Instances For
If x
is not an isolated point of X
, then x : OnePoint X
is not an isolated point
of OnePoint X
.
If X
is a non-compact space, then ∞
is not an isolated point of OnePoint X
.
If X
is not a compact space, then the natural embedding X → OnePoint X
has dense range.
Compactness and separation properties #
In this section we prove that OnePoint X
is a compact space; it is a T₀ (resp., T₁) space if
the original space satisfies the same separation axiom. If the original space is a locally compact
Hausdorff space, then OnePoint X
is a normal (hence, T₃ and Hausdorff) space.
Finally, if the original space X
is not compact and is a preconnected space, then
OnePoint X
is a connected space.
For any topological space X
, its one point compactification is a compact space.
The one point compactification of a T0Space
space is a T0Space
.
The one point compactification of a T1Space
space is a T1Space
.
The one point compactification of a weakly locally compact Hausdorff space is a T₄ (hence, Hausdorff and regular) topological space.
If X
is not a compact space, then OnePoint X
is a connected space.
If X
is an infinite type with discrete topology (e.g., ℕ
), then the identity map from
CofiniteTopology (OnePoint X)
to OnePoint X
is not continuous.
A concrete counterexample shows that Continuous.homeoOfEquivCompactToT2
cannot be generalized from T2Space
to T1Space
.
Let α = OnePoint ℕ
be the one-point compactification of ℕ
, and let β
be the same space
OnePoint ℕ
with the cofinite topology. Then α
is compact, β
is T1, and the identity map
id : α → β
is a continuous equivalence that is not a homeomorphism.