T2 Property is Hereditary

Theorem

Let $T = \left({S, \tau}\right)$ be a topological space which is a $T_2$ (Hausdorff) space.

Let $T_H = \left({H, \tau_H}\right)$, where $\varnothing \subset H \subseteq S$, be a subspace of $T$.

Then $T_H$ is a $T_2$ (Hausdorff) space.

Proof

Let $T = \left({S, \tau}\right)$ be a $T_2$ (Hausdorff) space.

Then:

$\forall x, y \in S, x \ne y: \exists U, V \in \tau: x \in U, y \in V: U \cap V = \varnothing$

That is, for any two distinct points $x, y \in S$ there exist disjoint open sets $U, V \in \vartheta$ containing $x$ and $y$ respectively.

We have that the set $\tau_H$ is defined as:

$\tau_H := \left\{{U \cap H: U \in \tau}\right\}$

Let $x, y \in H$ such that $x \ne y$.

Then as $x, y \in S$ we have that:

$\exists U, V \in \tau: x \in U, y \in V, U \cap V = \varnothing$

As $x, y \in H$ we have that:

$x \in U \cap H, y \in V \cap H: \left({U \cap H}\right) \cap \left({V \cap H}\right) = \varnothing$

and so the $T_2$ axiom is satisfied in $H$.

$\blacksquare$

Sources

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 2$: Functions, Products, and Subspaces