Can anyone will help me to get the concept of Sierpiński topology ? how Can we define Sierpiński topology over every set ?
2026-02-22 19:08:48.1771787328
what is Sierpiński topology?
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You might mean the Sierpiński topology instead. This is a topology on the set $\{0,1\}$ with one non-trivial open set, like $\{\emptyset, \{0,1\},\{0\}\}$, so exactly one open singleton and the other one closed. Whether we choose the $0$ to be isolated or $1$, is a matter of convention, as the results will always be homeomorphic.
It’s the minimal example of a space that is $T_0$ but not $T_1$.
So it’s a concrete space, not a class of topologies, as far as I’m aware.