Discrete Space is Non-Meager

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $T = \left({S, \tau}\right)$ be a discrete topological space.

Then $T$ is non-meager.


Proof 1

Let $U \subseteq S$ such that $U \ne \O$.

From Interior Equals Closure of Subset of Discrete Space, we have:

$U^\circ = U = U^-$

where $U^\circ$ is the interior and $U^-$ the closure of $U$.


So:

$\paren {U^-}^\circ = U \ne \O$

Thus, by definition, no non-empty subset of $S$ is nowhere dense.

So $S$ can not be the union (countable or otherwise) of nowhere dense subsets.

So by definition $S$ can not be meager.

Hence the result.

$\blacksquare$


Proof 2

Let $x \in S$ be an arbitrary point of $T$.

From Set in Discrete Topology is Clopen it follows that $\left\{{x}\right\}$ is open in $T$.

The result follows from Space with Open Point is Non-Meager.

$\blacksquare$