Discrete Space is Non-Meager
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Theorem
Let $T = \struct {S, \tau}$ 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 $\set x$ is open in $T$.
The result follows from Space with Open Point is Non-Meager.
$\blacksquare$