Hausdorff Space - Properties

Properties

Subspaces and products of Hausdorff spaces are Hausdorff, but quotient spaces of Hausdorff spaces need not be Hausdorff. In fact, every topological space can be realized as the quotient of some Hausdorff space.

Hausdorff spaces are T₁, meaning that all singletons are closed. Similarly, preregular spaces are R₀.

Another nice property of Hausdorff spaces is that compact sets are always closed. This may fail in non-Hausdorff spaces such as Sierpiński space.

The definition of a Hausdorff space says that points can be separated by neighborhoods. It turns out that this implies something which is seemingly stronger: in a Hausdorff space every pair of disjoint compact sets can also be separated by neighborhoods, in other words there is a neighborhood of one set and a neighborhood of the other, such that the two neighborhoods are disjoint. This is an example of the general rule that compact sets often behave like points.

Compactness conditions together with preregularity often imply stronger separation axioms. For example, any locally compact preregular space is completely regular. Compact preregular spaces are normal, meaning that they satisfy Urysohn's lemma and the Tietze extension theorem and have partitions of unity subordinate to locally finite open covers. The Hausdorff versions of these statements are: every locally compact Hausdorff space is Tychonoff, and every compact Hausdorff space is normal Hausdorff.

The following results are some technical properties regarding maps (continuous and otherwise) to and from Hausdorff spaces.

Let f : X → Y be a continuous function and suppose Y is Hausdorff. Then the graph of f, is a closed subset of X × Y.

Let f : X → Y be a function and let be its kernel regarded as a subspace of X × X.

• If f is continuous and Y is Hausdorff then ker(f) is closed.
• If f is an open surjection and ker(f) is closed then Y is Hausdorff.
• If f is a continuous, open surjection (i.e. an open quotient map) then Y is Hausdorff if and only if ker(f) is closed.

If f,g : X → Y are continuous maps and Y is Hausdorff then the equalizer is closed in X. It follows that if Y is Hausdorff and f and g agree on a dense subset of X then f = g. In other words, continuous functions into Hausdorff spaces are determined by their values on dense subsets.

Let f : X → Y be a closed surjection such that f−1(y) is compact for all y ∈ Y. Then if X is Hausdorff so is Y.

Let f : X → Y be a quotient map with X a compact Hausdorff space. Then the following are equivalent

• Y is Hausdorff
• f is a closed map
• ker(f) is closed