Countable Complement Space is not T3, T4 or T5

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $T = \left({S, \tau}\right)$ be a countable complement topology on an uncountable set $S$.


Then $T$ is not a $T_3$ space, $T_4$ space or $T_5$ space.


Proof

We have that a Countable Complement Space is a $T_1$ space.

From $T_1$ Space is $T_0$ Space, $T$ is a $T_0$ space.


We then have that a Countable Complement Space is not $T_2$.

From Regular Space is $T_2$ Space, $T$ is not a regular space.

By definition, a regular space is a space that is both a $T_0$ space and a $T_3$ space.

But $T$ is a $T_0$ space and not a regular space.

So it follows that $T$ cannot be a $T_3$ space.


Next we have that a Normal Space is a $T_3$ Space.

But as $T$ is not a $T_3$ space, $T$ cannot be a normal space.

By definition, a normal space is a space that is both a $T_1$ space and a $T_4$ space.

But $T$ is a $T_1$ space and not a normal space.

So it follows that $T$ cannot be a $T_4$ space.


Finally we have that a $T_5$ Space is a $T_4$ Space.

But as $T$ is not a $T_4$ space, $T$ cannot be a $T_5$ space.

$\blacksquare$


Sources