A question about the indescrete space If $($X$,\tau)$ is a topological space with the property that the only infinite subset of $X$ that is open is $X$ itself,is necessarily the indiscrete space?
Ibelieve it's not, because if $\alpha$ $\in$ $X$ then $X$ \{$\alpha$} is infinite so it is not in $\tau$ and is closed therefore {$\alpha$} is open and it's in $\tau$.Is this approach wright ?
Must finite sets be open in this topology?

I am assuming that the ``indiscrete'' space you refer to is one with the trivial topology $\tau = \{\emptyset, X\}$. If I am wrong, please clarify.
Let $x\in X$. Set $\tau= \{\emptyset, \{x\}, X\}$. This can be checked to be a topology, and satisfies your requirement.
