T2 Property is Hereditary

Theorem

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

Let $T_H = \struct {H, \tau_H}$, where $\O \subset H \subseteq S$, be a subspace of $T$.

That is, the property of being a $T_2$ (Hausdorff) space is hereditary.

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

Then:

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

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

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

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

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

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

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

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

$x \in U \cap H, y \in V \cap H: \paren {U \cap H} \cap \paren {V \cap H} = \O$

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

$\blacksquare$

1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $4$: The Hausdorff condition: $4.2$: Separation axioms: Proposition $4.2.4 \ \text{(a)}$
1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $4$: The Hausdorff condition: Exercise $4.3: 2$
1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $2$: Separation Axioms: Functions, Products, and Subspaces