Questions: Urysohn Metrization Theorem

Question 1

A topological space X is normal but not second-countable. What does the Urysohn Metrization Theorem tell you about X?

• A. X is metrizable, since normality alone is sufficient
• B. X is not metrizable, since normality alone is insufficient
• C. The theorem is silent — it gives no conclusion about X's metrizability
• D. X is metrizable only if it is also Hausdorff

Answer: (C)

The Urysohn Metrization Theorem states that second-countable AND normal implies metrizable. It is a sufficient condition requiring both hypotheses. If one hypothesis is missing, the theorem simply does not apply — it says nothing about whether X is or is not metrizable. A normal space without second-countability may or may not be metrizable; the theorem is silent. (The long line is a classic example of a normal, non-second-countable, non-metrizable space.)

Question 2

What does it mean to say that a topological space X is metrizable?

• A. X has a distance function, but it may generate a coarser topology than the one given
• B. X can be given a metric that generates a topology that agrees with the given topology
• C. X embeds into some metric space, possibly losing some open sets in the process
• D. Every continuous function on X extends to a metric space

Answer: (B)

Metrization is precise: a metric must exist on X such that the topology it generates is exactly the given topology — not merely related to it, but identical. A coarser or finer metric topology would not count. The key point is that an abstract topological space defined by open sets turns out to be secretly described by a distance function. Options A and C describe weaker conditions (the topology might not be recovered); option D is unrelated.

Question 3

In the proof of the Urysohn Metrization Theorem, normality is used to construct Urysohn functions. This means normality guarantees that for any two disjoint closed sets, there exists a continuous function separating them.

• T. True
• F. False

Answer: True

This is precisely the content of Urysohn's lemma: a space is normal if and only if for any two disjoint closed sets C and D, there is a continuous f : X → [0,1] with f = 0 on C and f = 1 on D. Normality is the exact topological condition that makes such separator functions exist, and the metrization proof assembles a countable family of these functions to build the embedding into ℓ²(ℕ).

Question 4

Most second-countable space is metrizable, regardless of whether it is normal.

• T. True
• F. False

Answer: False

Second-countability alone is not sufficient for metrizability. The Urysohn Metrization Theorem requires both second-countability and normality. A second-countable space that fails normality need not be metrizable — the two conditions work together, with second-countability providing a countable base for building the function family, and normality providing the Urysohn functions themselves. Dropping either condition breaks the argument.

Question 5

Explain why the Urysohn Metrization Theorem uses an embedding into ℓ²(ℕ), and what role second-countability plays in making this embedding possible.

Model answer: Second-countability guarantees a countable base {U₁, U₂, …}, which yields a countable collection of pairs (Uᵢ, Uⱼ) with cl(Uᵢ) ⊆ Uⱼ. Normality then produces a Urysohn function for each such pair. Because the collection of functions is countable, the map x ↦ (f₁(x), f₂(x), …) lands in ℓ²(ℕ), a separable Hilbert space with a natural metric. Pulling back the ℓ² metric gives the desired metric on X. Without second-countability, the function family might be uncountable, making the ℓ²(ℕ) embedding unavailable.

The embedding strategy — mapping X into a known metric space and pulling back the metric — requires that the coordinate functions form a countable sequence. Second-countability is exactly the condition that makes this countability available. This is why the theorem fails without it: you cannot assemble an uncountable family of functions into coordinates of ℓ²(ℕ). The two hypotheses are not redundant; they play complementary roles in the proof.