We proved some results relating the different countability properties we introduced in the previous lecture for general topological spaces and metrizable spaces.

First, we showed that being second-countable always implies being first-countable. In particular, if \(\mathcal{B}\) is a countable basis for a topological space \(X\), then for any point \(x \in X\), we have that \(\mathcal{B}_x = \{U \in \mathcal{B} \mid x \in U\}\) is a countable local basis at \(x\).

Next, we showed that being first-countable imples being separable. In particular, if \(\mathcal{B}\) is a countable basis for a topological space \(X\), assuming without loss of generalith that every \(U \in \mathcal{B}\) is nonempty, we can select a point \(x_U \in U\) for each such \(U\) where \(\{x_U \mid U \in \mathcal{B}\}\) is a countable dense subset of \(X\).

Moving on to metrizable spaces, recall that a topological space \(X\) is metrizable if and only if there is a metric \(d\) on \(X\) generating the topology. This means the collection of open balls \(\{B_d(x, r) \mid x \in X, \; r > 0\}\) forms a basis for the topology.

We showed that every metrizable space is first-countable. In particular, let \(d\) be a metric on \(X\) generating the topology for any point \(x \in X\), consider \(\mathcal{B}_x = \{B_d(x, \frac{1}{n}) \mid n \in \mathbb{Z}^{> 0}\}\). This forms a countable local basis at \(x\).

Finally, we showed that every separable metrizable space is second-countable. Suppose \(A \subseteq X\) is a countable dense set and consider \(\mathcal{B} = \{B_d(x, \frac{1}{n}) \mid n \in \mathbb{Z}^{>0}\}\). This forms a countable basis for the topology.