Lindelöf, Countably Compact, and BW Spaces Review

We will now review some of the recent material regarding Lindelöf spaces, countably compact spaces, and BW spaces.

• Recall from the Lindelöf and Countably Compact Topological Spaces page that a topological space $X$ is said to be Lindelöf if every open cover of $X$ has a countable subcover.

• Furthermore, $X$ is aid to be Countably Compact if every countable cover of $X$ has a finite subcover.

• On The Lindelöf Lemma page we proved a very nice theorem which says that every second countable topological space is Lindelöf.

• On the Bolzano Weierstrass Topological Spaces page we looked at a new type of topological space. We said that a topological space $X$ is a Bolzano Weierstrass Space or simply as "BW Space" if every infinite subset of $X$ has an accumulation point. We saw that $\mathbb{R}$ with the usual topology is not a BW space. This is because the infinite subset of integers $\mathbb{Z} \subset \mathbb{R}$ has no accumulation point.

• On the Hausdorff Spaces Are BW Spaces If and Only If They're Countably Compact page we saw that if $X$ is a Hausdorff space that $X$ is a BW space if and only if $X$ is countably compact.

• On the Compact Spaces as BW Spaces page we (more generally) saw that if $X$ is a compact space then $X$ is a BW space. Further restrictions need to be applied for the converse of this result to be true though.

• On the The Lebesgue Number Lemma page we looked at a very famous result known as the Lebesgue Number Lemma. It said that if $(X, d)$ is a metric space that is also a BW space that for every open cover $\mathcal F$ there exists an $\epsilon > 0$ such that for all $x \in X$ there exists a $U \in \mathcal F$ such that:

(1)

\begin{align} \quad B(x, \epsilon) \subseteq U \end{align}

• Such a number $\epsilon > 0$ satisfying the definition above is called a Lebesgue Number.

• Using this result, we saw on the Metric Spaces Are Compact Spaces If and Only If They're BW Spaces page that if $X$ is a metric space then $X$ is compact if and only if $X$ is a BW space.

• As a nice consequence, on the Metric Spaces Are Compact Spaces If and Only If They're Countably Compact page we further saw that if $X$ is a metric space then $X$ is compact if and only if $X$ is countably compact. Compactness implying countably compactness was obvious. The converse was rather simple to show too, since if $X$ is a metric space then $X$ is Hausdorff and countably compact which implies that $X$ is a BW metric spaces which implies that $X$ is compact.