Theorem 29.1 in Munkres's Topology Could someone please explain the highlighted sentence in this proof? I understand that $C$ is contained in $X$, but I don't understand why that implies it is a compact subspace of $X$ given that it is a compact subspace of $Y$. Thanks in advance.


If $X$ is a subspace of $Y$, the open sets of $X$ are of the form $X\cap U$ where $U$ is an open set in $Y$. Therefore any open covering of $C$ as a subset of $X$ is of the form $\{X\cap U_i\}_{i \in I}$. But then $\{U_i\}_{i \in I}$ is an open covering of $C$ in $Y$, so there is a finite sub-covering.
