I solved some short questions and I want to know if they are correct:
- Every subset of a topological space is either open or closed.
Answer: Wrong. Let $X\subseteq X$ is a subspace and $X$ is open and closed in $X$.
- An open set is not closed.
Answer: Wrong. $\emptyset$ is closed and open.
- Compact subsets of a Hausdorff space are closed.
Answer: True. $A\subset X$ compact, $X$ Hausdorff $\implies$ $A$ is closed in $X$
- If a subset of a topological space has an cover of open sets, then it's compact.
Answer: Wrong. $A\subset X$ is compact iff every cover of open sets has a finite subcover.
- Surfaces with the same Euler-characteristics are homeomorph.
Answer: Wrong. It is true that if $F_1\simeq F_2\implies\chi(F_1)=\chi(F_2)$ but not $\chi(F_1)=\chi(F_2)\implies F_1\simeq F_2 $.