Urysohn separation and a metrization construction

Try

In a metric space, disjoint closed sets can be separated by a continuous distance-ratio function. Urysohn’s lemma constructs a similar function in a normal space without any distance available.

The key is to place open neighborhoods between one closed set and the complement of the other, with closures nested at every intermediate dyadic level. The resulting function records the earliest level at which a point enters. Merely nesting the sets without controlling closures would not prove continuity.

Predict

Predict the endpoint values. The first closed set A should receive zero and the second B should receive one. Intermediate values are chosen through neighborhoods rather than an explicit algebraic distance.

Predict why a countable family of such functions might define a metric. A weighted sum of coordinate differences is finite, but it becomes a genuine metric only if the functions separate points. To recover the original topology, they must also distinguish points from neighborhood complements.

Experiment

Track the dyadic construction.

Stage	New levels
Initial	0 and 1
First insertion	1/2
Second insertion	1/4 and 3/4
Later insertions	All dyadic rationals in [0,1]

At every stage maintain closure(U_r)⊆U_s whenever r<s. Normal shrinking supplies each new intermediate neighborhood.

Understand

Set U₁=X without B and choose U₀ with A⊆U₀⊆closure(U₀)⊆U₁. Repeated normal shrinking inserts the dyadic neighborhoods. Define f(x) as the infimum of the dyadic r with x∈U_r, adding the value one to the infimum set so it is never empty. Then f=0 on A and f=1 on B.

For 0<t<1, the strict sublevel set is the union of U_r for r<t. The strict superlevel set is the union of X without closure(U_r) for t<r<1. For the nontrivial implication, if f(x)>t choose t<r<s<f(x); then x is outside U_s and hence outside closure(U_r). Conversely being outside closure(U_r) prevents membership in any earlier U_q, giving f(x)≥r>t. Both level sets are open, proving continuity. Endpoint cases follow from the range [0,1]. This proves Urysohn’s lemma.

For metrization, a regular second-countable T₁ space is normal. Choose countable open covers U_n of A and V_n of B with closures missing the opposite closed set, using regularity and Lindelöfness. Remove the first n opposite closures from each nth open before taking unions. The resulting open unions still contain A and B, and comparing the two indices proves disjointness.

Choose a countable basis. For every basis pair with closure(B_i)⊆B_j, Urysohn supplies a function one on closure(B_i) and zero outside B_j. These countably many functions separate points and neighborhoods: shrink a point’s neighborhood twice and choose suitable basis members between the closures. Evaluation embeds X into the countable cube. The metric Σ2^(−n)|u_n−v_n| induces its product topology, so its pullback metrizes X. This gives the construction underlying Urysohn’s metrization theorem.

Apply

Verify the weighted metric’s topology. A small distance controls any fixed finite list of coordinates because their weights are positive. Conversely, choose a long finite initial segment so the remaining weight sum is small, then restrict those initial coordinates enough to make the total distance small.

Repair “Point separation alone proves an embedding.” It proves injectivity. The neighborhood-separation property is what makes the inverse onto the image continuous and ensures the pulled-back metric has the original topology.

Continue

Tietze extension next uses Urysohn functions repeatedly to correct an arbitrary function on a closed subset. Optional depth: write the two nested regular shrinkings needed to find basis members with x∈B_i, closure(B_i)⊆B_j⊆U. This fills the local containment step in the countable-function embedding argument.