If $X$ is Hausdorff space and $x \in X$ isn't isolated point then $X-\{x\}$ isn't compact.
Well, I am trying to show that if $X-\{x\}$ is compact then there is a contradiction.
Ok, I know that every compact subset of a Hausdorff space is closed. Suppose that $X - \{x\}$ is compact, then $X - \{x\}$ is closed so $\{x\}$ is open set, but he isn't isolated and it arrives in a contradiction?