Indiscrete Space is Non-Meager

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Theorem

Let $T = \left({S, \left\{{\varnothing, S}\right\}}\right)$ be an indiscrete topological space.

Then $T$ is non-meager.


Proof

Let $H \subseteq S$ such that $H \ne \varnothing$.

From Limit Points of Indiscrete Space, every point of $S$ is a limit point of $H$.

So, by definition:

$H^- = S$

where $H^-$ is the closure of $H$ in $T$.

Now the interior of $S$ is $S$ itself (trivially, by definition).

So:

$\left({H^-}\right)^\circ = S \ne \varnothing$

where $\left({H^-}\right)^\circ$ is the interior of the closure of $H$.

So, by definition, $H$ is not nowhere dense.

(Note that if $H = \varnothing$, then Empty Set is Nowhere Dense applies, which is why the stipulation above that $H \ne \varnothing$.)

It follows that $H$ can not be the countable union of nowhere dense subsets of $S$.

So $H$ is, by definition, not meager, so must be non-meager.

$\blacksquare$


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