T5 Property is Hereditary

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Theorem

Let $T = \struct{S, \tau}$ be a topological space which is a $T_5$ space.

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


Then $T_H$ is a $T_5$ space.


Proof

Let $T = \struct{S, \tau}$ be a $T_5$ space.

Then:

$\forall A, B \subseteq S, \map {\operatorname{cl}_S} A \cap B = A \cap \map {\operatorname{cl}_S} B = \O: \exists U, V \in \tau: A \subseteq U, B \subseteq V, U \cap V = \O$

where $\map {\operatorname{cl}_S} A$ denotes the closure of $A$ in $S$.

That is:

For any two separated sets $A, B \subseteq S$ there exist disjoint open sets $U, V \in \tau$ containing $A$ and $B$ respectively.


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

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


Let $A, B \subseteq H$ such that $\map {\operatorname{cl}_H} A \cap B = A \cap \map {\operatorname{cl}_H} B = \O$.

That is, $A$ and $B$ are separated in $H$.


Then:

\(\displaystyle \map {\operatorname{cl}_S } A \cap B\) \(=\) \(\displaystyle \map {\operatorname{cl}_S } A \cap \paren{H \cap B}\) as $B \subseteq H$
\(\displaystyle \) \(=\) \(\displaystyle \paren{\map {\operatorname{cl}_S } A \cap H } \cap B\) Intersection is Associative
\(\displaystyle \) \(=\) \(\displaystyle \map {\operatorname{cl}_H } A \cap B\) Closure in Subspace
\(\displaystyle \) \(=\) \(\displaystyle \O\) Assumption

Similarly:

$A \cap \map {\operatorname{cl}_S} B = \O$

So $A$ and $B$ are separated in $S$.


Because $T$ is a $T_5$ space, we have that:

$\exists U, V \in \tau: A \subseteq U, B \subseteq V, U \cap V = \O$

It follows that:

$\exists U \cap H, V \cap H \in \tau_H : A \subseteq U \cap H, B \subseteq V \cap H, \paren{U \cap H} \cap \paren{V \cap H} = \O$

and so the $T_5$ axiom is satisfied in $H$.

$\blacksquare$


Sources