Either-Or Topology is Non-Meager

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Theorem

Let $T = \left({S, \tau}\right)$ be the either-or space.


Then $T$ is non-meager.


Proof 1

From the definition of the either-or space, we have that every point $x$ in $T$ (apart from $0$) forms an open set of $T$.

So every non-empty subset of $T$ (apart from $\left\{{0}\right\})$ contains at least one open set of $T$.

So no subset of $T$ is nowhere dense in $T$.

So $T$ is not a countable union of subsets of $S$ which are nowhere dense in $T$.

Hence the result by definition of non-meager.

$\blacksquare$


Proof 2

From the definition of the either-or space, we have that every point $x$ in $T$ (apart from $0$) forms an open set of $T$.

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

$\blacksquare$


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