Lindelöf, Countably Compact, and BW Spaces Review

# 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:

\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.