Let $(X,\tau)$ be a topological space. Prove that $\tau$ is the finite-closed topology on $X$ if and only if (i)$(X,\tau)$ is a $T_1$-space, and (ii) every infinite subset of $X$ is dense in $X$.
I already proved the forward direction but I'm stuck on the backward direction. We know that everyone singleton is closed because of i) and from ii) every open set intersects any infinite set non trivially. Now I need to figure out how to show that every open set are just infinite sets with countably finite points removed, thus the topology is finite-closed.
Best Answer
Let $U$ be a non-empty open set. If $X \setminus U$ has an infinite number of points then the infinite set $X \setminus U$ cannot be dense. This is because it does not intersect $U$. This contradiction proves that $X \setminus U$ is finite whenever $U$ is a non-empty open set.