KindiakMath

The Hunt for Metrizability

The crucial property that gives us massive control over a topological space is its compactness. For metrizable spaces, compactness is equivalent to sequential compactness. This therefore raises a crucial question: when is a topological space metrizable?

We first make some observations on the necessary conditions for a space to be metrizable.

Lemma 1. If K is a metrizable space, then it is first-countable, i.e. for any x \in K, there exists a countable basis \mathcal B_x of neighbourhoods of x.

Proof. Define \mathcal B_x := \{ B_d(x, r) : r \in \mathbb Q^+ \}, where d is the assumed metric underlying K.

Rather than having a countable basis for just near any fixed x, it is sometimes useful to have a countable basis for the whole topological space. This is known as second-countability.

Definition 1. A topological space K is second-countable if it is generated by a countable basis.

Lemma 2. Second-countable spaces are first-countable.

Proof. If K is second-countable with countable basis \mathcal B, then for any x \in K, \mathcal B_x := \{B \in \mathcal B : x \in B\} yields a countable basis near x.

Example 1. \mathbb R^n is second-countable.

Proof. Assuming the standard metric, since \overline{\mathbb Q} = \mathbb R, \mathbb R^n has the countable basis

\{B_d(\mathbf x, r) : \mathbf x \in \mathbb Q^n, r \in \mathbb Q_{\geq 0}\}.

This fact also demonstrates that \overline{\mathbb Q^n} = \mathbb R^n.

We might think, therefore, that there might be some connection between second-countability and metrizability, and even first-countability. None of these notions are identical to each other, and we prove them using several counterexamples.

Example 2. Equip the set K := \{0, 1\} with the topology \mathcal T := \{ \emptyset, \{ 0 \}, K \}. Then K is first-countable but not metrizable.

Proof. First-countability is obvious, and non-metrizability follows from non-Hausdorffness.

Example 3. Consider \mathbb R equipped with the discrete metric. Then any basis must contain the uncountable set \{\{x\} : x \in \mathbb R\}.

Example 2 kills two birds with one stone, exhibiting a metrizable space, and thus a first-countable space, that are both not second-countable. Can we find a second-countable space that is not metrizable?

This question remains elusive, and will require us to consider the separability of the topological space. We have discussed one type of separability—Hausdorffness. It’s relatives we will discuss in the next post.

—Joel Kindiak, 16 Apr 25, 2255H

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