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A topology may be presented only through open sets, with no numerical distance in sight. Urysohn’s metrisation theorem gives a route back to a metric: regularity supplies continuous functions that separate points from closed sets, while a countable base reduces the required family to countably many coordinates. The proof is a construction, not a slogan that every nice-looking space is metrizable.
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Chapter contents
Choose the route closest to the decision, or read from Chapter 1. The chapters remain open; these grouped routes are a navigable index.
Route 1 · Define the decision
Route 2 · Work the ledgers
Route 3 · Transfer the method
Route 4 · Check limits and evidence
Route 5 · Ask and continue
CHAPTER 1 OF 12 · DEFINE THE DECISION
1. State the theorem with conventions visible
One standard form says that every regular second-countable space is metrizable, where “regular” includes the T1 separation condition. Texts that separate T1 from regularity state the hypotheses accordingly.
Metrizable means there exists some metric whose open balls generate exactly the original topology. The theorem need not recover a pre-existing geometric distance.
CHAPTER 2 OF 12 · DEFINE THE DECISION
2. Identify the two resources in the proof
Regularity lets a point and a disjoint closed set be separated by neighbourhoods. Together with normality consequences available under second countability, Urysohn-type functions take values in [0,1] and separate selected closed sets.
Second countability supplies a countable base. Countability is what lets the separating functions be listed as f1,f2,… rather than indexed by an uncontrollable family.
CHAPTER 3 OF 12 · WORK THE LEDGERS
3. Build the countable pair ledger
Let {Bn} be a countable base. Consider ordered pairs (Bi,Bj) whose closures satisfy cl(Bi)⊂Bj. For each useful pair, choose a continuous function that is 1 on cl(Bi) and 0 outside Bj.
| Base pair | Required containment | Continuous coordinate role |
|---|---|---|
| (Bi,Bj) | cl(Bi)⊂Bj | Marks points deep inside Bi |
| Point x in Bi | Choose Bj around x | Gives a coordinate large at x |
| Point y outside Bj | Closed-side separation | Gives the same coordinate value 0 at y |
Because the base is countable, the useful ordered pairs are countable. Enumerate the resulting functions.
CHAPTER 4 OF 12 · WORK THE LEDGERS
4. Map the space into the countable cube
Define F(x)=(f1(x),f2(x),…) in [0,1]^N. Every coordinate is continuous, so F is continuous for the product topology.
| Property to prove | Coordinate argument | Result |
|---|---|---|
| Injective | Some fn separates distinct x and y | F(x)≠F(y) |
| Continuity | Each coordinate fn is continuous | F is continuous |
| Topology recovered | Coordinates refine each basic neighbourhood | F is an embedding |
| Metrizability | Countable cube has a product metric | Pull the metric back along F |
A common product metric is d(u,v)=Σ2^(−n)|un−vn|. Its weighted tail converges, and pulling it back gives dX(x,y)=d(F(x),F(y)).
CHAPTER 5 OF 12 · WORK THE LEDGERS
5. Check the metric rather than admiring the formula
Non-negativity, symmetry and the triangle inequality come from the cube metric. Separation follows from injectivity: if x≠y, some coordinate differs, so the distance is positive.
The decisive step is topological equivalence. Coordinate continuity gives one inclusion; the base-pair construction gives the reverse by finding finitely controlled coordinate neighbourhoods inside any chosen basic open set.
CHAPTER 6 OF 12 · TRANSFER THE METHOD
6. Work an example that already has a metric
For the real line with its usual topology, intervals with rational endpoints form a countable base and the space is regular. The theorem guarantees a compatible metric, although the familiar |x−y| already works.
This example checks the hypotheses but does not display the theorem’s full force. The construction matters most when the topology arrives abstractly rather than from an obvious formula.
CHAPTER 7 OF 12 · TRANSFER THE METHOD
7. Transfer to a countable product
A countable product of second-countable spaces is second countable: finite-coordinate basic opens can be chosen from countable bases, and there are only countably many such finite specifications.
If the factors are regular under the required separation conventions, the product route supports metrizability. An uncountable product may fail second countability even though every factor is metrizable.
CHAPTER 8 OF 12 · TRANSFER THE METHOD
8. Connect Urysohn functions and Tietze extension
Use Urysohn’s lemma and separating closed sets for the function-building engine. Use Tietze extension when the task is to extend an already defined continuous function from a closed subspace.
The roles differ: metrisation needs enough coordinates to encode topology; extension preserves a function while enlarging its domain.
CHAPTER 9 OF 12 · CHECK LIMITS AND EVIDENCE
9. Know why the hypotheses are not biconditional as stated
Every second-countable regular space is metrizable, but not every metrizable space is second countable. An uncountable discrete space has the discrete metric and no countable base.
Dropping separation can also fail. A pseudometric may identify distinct points, so a formula that generates a quotient-like topology does not automatically give a metric on the original set.
CHAPTER 10 OF 12 · CHECK LIMITS AND EVIDENCE
10. Use a proof and source checklist
State the separation convention, exhibit the countable base, show how base pairs are enumerated, justify the separating functions, prove F is an embedding and name a metric on the countable cube.
Romy Sharifi’s UCLA point-set topology notes give the embedding formulation and proof. MIT OpenCourseWare topology notes place Urysohn’s lemma, metrisation and Tietze extension in sequence.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Frequently asked questions
Does the theorem produce a unique metric?
No. Many different metrics can generate the same topology.
Is second countability necessary for every metrizable space?
No. It is a sufficient hypothesis here, not a property of all metric spaces.
Why use [0,1]^N?
It is a metrizable countable product whose coordinates can record the separating functions.
CHAPTER 12 OF 12 · ASK AND CONTINUE
12. Continue with a changed-hypothesis check
Test an uncountable discrete space. It is regular and metrizable, but not second countable. Explain why this refutes the converse without contradicting the theorem.
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