The OnePoint Compactification #
We construct the OnePoint compactification (the one-point compactification) of an arbitrary
X and prove some properties inherited from
Main definitions #
OnePoint: the OnePoint compactification, we use coercion for the canonical embedding
X → OnePoint X; when
Xis already compact, the compactification adds an isolated point to the space.
OnePoint.infty: the extra point
Main results #
- The topological structure of
- The connectedness of
OnePoint Xfor a noncompact, preconnected
T₀for a T₀ space
T₁for a T₁ space
OnePoint Xis normal if
Xis a locally compact Hausdorff space
one-point compactification, compactness
Definition and basic properties #
We define a topological space structure on
OnePoint X so that
s is open if and only if
(↑) ⁻¹' sis open in
∞ ∈ s, then
((↑) ⁻¹' s)ᶜis 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
(↑) has dense range, so it is a dense embedding.
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.
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.