Open and Closed Sets in Indiscrete Topology

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Theorem

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

Let $H \subseteq S$.


Open Sets in Indiscrete Topology

$H$ is an open set of $T$ if and only if either $H = S$ or $H = \varnothing$.


Closed Sets in Indiscrete Topology

$H$ is a closed set of $T$ if and only if either $H = S$ or $H = \varnothing$.


$F_\sigma$ Sets in Indiscrete Topology

$H$ is an open set of $T$ if and only if either $H = S$ or $H = \varnothing$.


$G_\delta$ Sets in Indiscrete Topology

$H$ is a $G_\delta$ (G-delta) set of $T$ if and only if either $H = S$ or $H = \O$.


Sources