# If every non-empty open set of $X$ is dense in $X$, then if $C$ is an arbitrary own closed of $X$, then interior part of $C$ is empty.

Let $$X$$ a topological space. I have to prove it: If every non-empty open set of $$X$$ is dense in $$X$$, then if $$C$$ is an arbitrary own closed of $$X$$, then interior part of $$C$$ is empty. I tried to prove it but I don’t arrive at corrected conclusion.

• What if $C=X$ ? – TheSilverDoe Apr 3 at 18:53

If interior of $$C$$ is non empty, then it's a non-empty open set, then it is dense, then closure of $$C$$ is $$X$$, then closure of $$C$$ is $$X$$, then $$C = X$$ (as closed subset is closure of itself).