It follows that if two subsets and are separated by a function then so are their closures.
Also it follows that if two subsets and are separated by a function then and are separated by neighbourhoods.
A normal space is a topological space in which any two disjoint closed sets can be separated by neighbourhoods. Urysohn's lemma states that a topological space is normal if and only if any two disjoint closed sets can be separated by a continuous function.
Urysohn's lemma has led to the formulation of other topological properties such as the 'Tychonoff property' and 'completely Hausdorff spaces'. For example, a corollary of the lemma is that normal T1 spaces are Tychonoff.
The procedure is an entirely straightforward application of the definition of normality (once one draws some figures representing the first few steps in the induction described below to see what is going on), beginning with two disjoint closed sets. The clever part of the proof is the indexing of the open sets thus constructed by dyadic fractions.
Once we have these sets, we define if for any ; otherwise for every where denotes the infimum. Using the fact that the dyadic rationals are dense, it is then not too hard to show that is continuous and has the property and
In order to construct the sets we actually do a little bit more: we construct sets and such that
and for all
and are open and disjoint for all
For is contained in the complement of and the complement of is contained in
Since the complement of is closed and contains the latter condition then implies condition (2) from above.
This construction proceeds by mathematical induction. First define and Since is normal, we can find two disjoint open sets and which contain and respectively. Now assume that and the sets and have already been constructed for Since is normal, for any we can find two disjoint open sets which contain and respectively. Call these two open sets and and verify the above three conditions.