Importance of normality

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Introduction

The word normal may suggest meanings of being usual, or ordinary. However, normality as a property of topological spaces is not the most ordinary of properties. It has a number of good and powerful features, but it also has its limitations, which open the way for a number of other properties.

Usefulness of the normality assumption

Ability to separate closed subsets

In a normal space, any two disjoint closed subset are separated by disjoint open subsets; this is a useful assumption and has many applications. One application is to algebraic topology; we can use excision to determine the homology of the complement of the union of two disjoint closed subsets, using the homology of each of them. There are a number of other situations where the assumption comes useful.

Ability to separate by continuous functions

Urysohn's lemma shows that any two disjoint closed subsets in a normal space can be separated by a continuous function to [0,1] which takes the value 0 on one closed set and 1 on the other. This is an extremely useful result; for instance, it shows that any connected normal space must have cardinality at least that of the continuum.

The ability to pass from the definition involving disjoint open subsets to the definition involving continuous functions, arises from the fact that closed subsets are the complements of open sets, so we can induct (unlike in th ecase of regular and Hausdorff spaces, where one of the sides is a point, so an induction is impossible).

Complete regularity

In particular, normal spaces are completely regular, and the assumption of complete regularity is extraordinarily useful. Two main uses:

  • Any completely regular space embeds as a dense subspace of a compact Hausdorff space. Thus, so does every normal space.
  • Any completely regular space occurs as the underlying space of a uniform space. Thus, so does every normal space.

Tietze extension theorem

Further information: Tietze extension theorem

Perhaps the most useful result about normal spaces is the Tietze extension theorem, which allows us to extend a continuous function from any closed subset to [0,1], to a function on the whole space. Tietze extension allows us to define a function step-by-step; first define it on a closed subset, then define it on the rest of the space. It is a crucial ingredient in proving that CW-complexes are normal: the fact we use is that the closed unit disc is a normal space.

Ubiquity of normality

Metrizable spaces

Further information: metrizable implies normal

All metrizable spaces are normal. Metrizable spaces constitute a large class of spaces; they include all sub-Euclidean spaces, all manifolds.

In fact, more generally, any ordered field-metrizable space is normal.

CW-spaces

Further information: CW implies normal

Any [{CW-space]], viz., any space which admits the structure of a CW-complex, is normal.

Linearly orderable spaces

Further information: linearly orderable implies normal

Any linearly orderable space, viz., any topological space which occurs as the underlying space of a linearly ordered space, is normal.

Non-closed nature of normality

Unlike the lower separation axioms, normality suffers in not being closed under taking either subspace or products. Normality differs from lower separation axioms in having closed sets on both sides of the separation. The advantages of this have already been seen: we are able to use separation by open sets to obtain separation by continuous functions, because closed sets are the complements of open sets.

The disadvantage is that while points in a subspace are the same thing as points inside the whole space, and points inside a product are tuples of points inside individual spaces, no such simple description is possible for closed subsets. A subspace can have closed subsets which are far from closed inside the whole space, and two disjoint closed subsets inside a subspace, may have an intersection between their closures in the whole space. Similarly, a product can have closed subsets which are very complicated in appearance and are far from closed rectangles.

Thus, a number of variations on the property of normality have been explored, which remain preserved on taking subspaces and products. Further information: varying normality