T3 1/2 Property is Hereditary

Theorem

Let $T = \struct {S, \tau}$ be a topological space which is a $T_{3 \frac 1 2}$ space.

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

Let $T = \struct {S, \tau}$ be a $T_{3 \frac 1 2}$ space.

Then:

For any closed set $F \subseteq S$ and any point $y \in S$ such that $y \notin F$, there exists an Urysohn function for $F$ and $\set y$.

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

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

Let $F \subseteq H$ such that $F$ is closed in $H$.

Let $y \in H$ such that $y \notin F$.

Because $T$ is a $T_{3 \frac 1 2}$ space, we have that there exists an Urysohn function for $F$ and $\set y$:

That is, there exists a continuous mapping $f: S \to \closedint 0 1$, where $\closedint 0 1$ is the closed unit interval, such that:

$f {\restriction_F} = 0, f {\restriction_{\set y} } = 1$

where $f {\restriction_F}$ denotes the restriction of $f$ to $F$.

That is:

$\forall a \in F: \map f a = 0$

$\forall b \in \set y: \map f b = 1$

From Continuity of Composite with Inclusion, as $f$ is continuous on $S$, then $f {\restriction_H}$ is continuous on $H$.

Thus $f {\restriction_H}$ is an Urysohn function for $F$ and $\set y$ in $H$.

So the $T_{3 \frac 1 2}$ axiom is satisfied in $H$.

$\blacksquare$

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