Condition for Closed Extension Space to be T5 Space

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Theorem

Let $T = \struct {S, \tau}$ be a topological space.

Let $T^*_p = \struct {S^*_p, \tau^*_p}$ be the closed extension space of $T$.


Then:

$T^*_p$ is a $T_5$ space if and only if $T$ is a $T_5$ space vacuously

and $T^*_p$ in this case is also a $T_5$ space vacuously.


Proof

Sufficient Condition

Let $T^*_p$ be a $T_5$ space.

Then for any two separated sets $A, B \subseteq S$ there exist disjoint open sets $U, V \in \tau^*_p$ containing $A$ and $B$ respectively.

However, for any non-empty set $U \in \tau^*_p$, $p \in U$.

Hence no non-empty open sets in $T^*_p$ are separated.

Therefore $T^*_p$ is a $T_5$ space vacuously: there do not exist two separated sets.

$\Box$


Necessary Condition

Let $A, B \in \tau^*_p$.

By Closure of Open Set of Closed Extension Space, we have that:

$\forall U \in \tau^*_p: U \ne \O \implies U^- = S$

where $U^-$ is the closure of $U$.

Thus for all $A, B \ne \O$ it cannot be the case that $A^- \cap B = A \cap B^- = \O$.

Therefore $T^*_p$ does not contain two separated sets if and only if $T$ does not contain two separated sets.

Hence $T^*_p$ is a $T_5$ space vacuously if and only if $T$ is a $T_5$ space vacuously.

$\blacksquare$


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Sources

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