Indiscrete Non-Singleton Space is not T0

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $T = \left({S, \left\{{\varnothing, S}\right\}}\right)$ be an indiscrete topological space which has more than one element.

Then $T$ is not a $T_0$ (Kolmogorov) space.


Proof

Let $a, b \in S$.

By definition of indiscrete space, $S$ is the only open set in $T$ which is non-empty.

So (trivially) there is no open set in $T$ containing $a$ and not $b$, or $b$ and not $a$.

Hence the result, by definition of $T_0$ (Kolmogorov) space.

$\blacksquare$


Sources