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The convention topology definitionis: let $X$ be a set and let $\mathcal{T}$ be a family of subsets of $X$. Then $\mathcal{T}$ is called a topology on $X$ if:

  • Both the $\emptyset$ and $X$ are elements of $\mathcal{T}$.
  • Any union of elements of $\mathcal{T}$ is an element of $\mathcal{T}$.
  • Any intersection of finitely many elements of $\mathcal{T}$ is an element of $\mathcal{T}$.

My question is, if adding one extra condition:

  • Every element $i$ of $\mathcal{T}$ also has it's complement $X \setminus i$ in $\mathcal{T}$.

Is there a term for topologies with this condition ?

peng yu
  • 1,271
  • Such collection $\tau$ is the collection of all clopen subsets of $X$ . If $(X,\tau)$ is connected then $\tau$ is trivial /indiscrete topology on $X$. – Sourav Ghosh Nov 04 '21 at 03:09
  • It’s not necessary connected, more of a intersection of sigma algebra and topology – peng yu Nov 04 '21 at 03:13

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