Download PDF

Hausdorff space∗
yark†
2013-03-21 13:26:27
A topological space (X, τ ) is said to be T2 (or said to satisfy the T2 axiom)
if given distinct x, y ∈ X, there exist disjoint open sets U, V ∈ τ (that is,
U ∩ V = ∅) such that x ∈ U and y ∈ V .
A T2 space is also known as a Hausdorff space. A Hausdorff topology for a
set X is a topology τ such that (X, τ ) is a Hausdorff space.
Properties
The following properties are equivalent:
1. X is a Hausdorff space.
2. The set
∆ = {(x, y) ∈ X × X : x = y}
is closed in the product topology of X × X.
3. For all x ∈ X, we have
\
{x} = {A : A ⊆ X closed, ∃ open set U such that x ∈ U ⊆ A}.
Important examples of Hausdorff spaces are metric spaces, manifolds, and
topological vector spaces.
∗ hHausdorffSpacei
created: h2013-03-21i by: hyarki version: h31855i Privacy setting:
h1i hDefinitioni h54D10i
† This text is available under the Creative Commons Attribution/Share-Alike License 3.0.
You can reuse this document or portions thereof only if you do so under terms that are
compatible with the CC-BY-SA license.
1