Theorem 29.1 in Munkres's Topology

Question

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.

Answer 1

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.

Answer 2

Compactness can be defined as a space or as a subset.

As a space: $X$ is compact if every cover of $X$ by sets open in $X$ has a finite subcover.

As a subset: $X\subset Y$ is a compact subset if every cover of $X$ by sets open in $Y$ has a finite subcover.

Notice that, in the second definition, the open sets covering $X$ need not be contained in $X$. However, these two definitions are equivalent: for any topological space $Y$, a subset $X\subset Y$ is compact as a subset if and only if $X$ is compact as a space when viewed with the subset topology.

I won't prove this, because I think it is a good exercise. But if clarification is needeed, let me know.