T1 Property is Hereditary

Theorem

Let $T = \struct {S, \tau}$ be a topological space which is a $T_1$ (Fréchet) space.

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

Then $T_H$ is a $T_1$ (Fréchet) space.

Proof

Let $T$ be a $T_1$ (Fréchet) space.

That is:

$\forall x, y \in S$ such that $x \ne y$, both:

$\exists U \in \tau: x \in U, y \notin U$

and:

$\exists U \in \tau: y \in U, x \notin U$

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 $x, y \in S$:

$\exists U \in \tau: x \in U, y \notin U$

and:

$\exists U \in \tau: y \in U, x \notin U$

Then both:

$U \cap H \in \tau_H: x \in U \cap H, y \notin U \cap H$

and:

$U \cap H \in \tau_H: y \in U \cap H, x \notin U \cap H$

and so the $T_1$ axiom is satisfied.

$\blacksquare$

Sources

• 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