I am reading a book Hausdorff Compactifications and I don´t understand the sentence in bold. It is part of proof of the Theorem above (claim 1). I dont undersand in particular, what is meant by "augmenting" the open cover with open set.
Thank you for your help!
Theorem 1.11 from Chandler´s book:
- Closed subsets of compact spaces are compact.
- Compact subsets of
Hausdorff spaces are closed. - If $f : X \rightarrow Y$ is continuous
and $X$ is compact, then $f(X)$ is compact. - If $f : X \rightarrow Y$
is one-to-one and continuous, $X$ is compact, and $Y$ is Hausdorff
then $f$ is a homeomorphism onto $f(X)$.
Proof of 1.
Choose a compact space $X$ and its closed subset $F$. Let $\{\mathcal{O}_\alpha \}_{\alpha \in A}$ be an open cover of $F$.
We augment $\{\mathcal{O}_\alpha \}_{\alpha \in A}$ with the open set $ X \setminus F$ to obtain an open covering of the compact set $X$.
Best Answer
The author starts with an open cover $\{O_\alpha\mid\alpha\in A\}$ of $F$ and then adds to it the set $X\setminus F$. Since $F$ is closed, $X\setminus F$ is an open set. Therefore, not only is $\{X\setminus F\}\cup\{O_\alpha\mid\alpha\in A\}$ an open cover of $F$, but also an open cover of $X$, since $(X\setminus F)\cup\bigcup_{\alpha\in A}O_\alpha\supset(X\setminus F)\cup F=X$.