T4 and T3 Space is T 3 1/2

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Theorem

Let $T = \left({S, \tau}\right)$ be:

a $T_4$ space

and also:

a $T_3$ space.


Then $T$ is also a $T_{3 \frac 1 2}$ space.


Proof

Let $T = \left({S, \tau}\right)$ be a $T_4$ space which is also a $T_3$ space.


From it being $T_3$:

$\forall F \subseteq S: \complement_S \left({F}\right) \in \tau, y \in \complement_S \left({F}\right): \exists U, V \in \tau: F \subseteq U, y \in V: U \cap V = \varnothing$

Consider this $U \in \tau$, which is disjoint from $\left\{{y}\right\}$.

Then $\complement_S \left({U}\right)$ is a closed set which is disjoint from $F$ but such that $\left\{{y}\right\} \subseteq \complement_S \left({U}\right)$.


As $T$ is a $T_4$ space, we have that from Urysohn's Lemma there exists an Urysohn function $f$ for $F$ and $\complement_S \left({U}\right)$.

As $\left\{{y}\right\} \subseteq \complement_S \left({U}\right)$, this function $f$ is a Urysohn function for $F$ and $\left\{{y}\right\}$ as well.


So:

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 $\left\{{y}\right\}$.

which is precisely the definition of a $T_{3 \frac 1 2}$ space.

$\blacksquare$


Sources