what i had done :let X be a infinite set and there be open sets $A,B$ on it. such that $A\cap B =\varnothing$ or $A\cap B\ne\varnothing$then $X-A\cap X-B \ne \varnothing $ .which means there doesn't exist $X-A\cap X-B = \varnothing $ ,which implies that there is no separation of X in finite complement topology. is this proof correct.
[Math] Show that $X $ is a infinite set ,it is connected in the finite complement topology
general-topologyproof-verification
Best Answer
If $A \subset X$ is clopen in the cofinite topology, then ($A = \varnothing$ or $X - A$ is finite) and ($X - A = \varnothing$ or $A$ is finite).
Studying the four cases, we find only two possibilities: $A = \varnothing$ or $A = X$. This means that the only clopen sets are $\varnothing$ and $X$.