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Discrete topology is defined as: $P(X)=(A:A\subset X)$ Let X be an arbitrary non empty set, $T\subset P(X)$. It should also follow three axioms of union, intersection and $\phi$ and X should be in the $T$.

Q1) Does discrete topology has to have all sets as open?

Q2) I was trying to understand, how this discrete topology is the finest/strongest form of topology and all the rest of the topologies lie in between discrete and $\phi$. We can even define a topological space which includes open and closed sets, would not this be more finer than discrete topology.

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Q1) yes, it already contains all sets, which are therefore open.

Q2) everything is open, and everything is closed! this means for instance, that every map to another space is continous. This means also that every point is a connected component. you can think of further examples.

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  • $\begingroup$ This raises a doubt for me. Can a closed set $A \subset X$ be a part of (X,$T$) discrete topology $\endgroup$
    – vikram
    Sep 27, 2014 at 15:34
  • $\begingroup$ any set $A \subset X$ is part of the discrete topology. Hence any set $A$ is open. But also $X-A$ is a subset, hence also open. Therefore $A$ is as complement of its own open complement closed. $\endgroup$
    – Daniel Valenzuela
    Sep 27, 2014 at 15:37

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