Is every finite, nondiscrete $T_0$ space connected? What I've tried is to find a separation and thus get a contradiction with the statement that the space is nondiscrete. After some time I've got lost in it. It seems so "picky".
2026-03-29 14:57:42.1774796262
Is every finite, nondiscrete $T_0$ space connected?
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Let $X=\{1,2,3\}$ be endowed with topology $\tau=\{\varnothing,\{2\},\{3\},\{1,2\},\{2,3\},\{1,2,3\}\}$.
Then $X$ is $T_0$ and e.g. $\{3\}$ is a non-trivial clopen set.