Hausdorff space

For a topological space ${\displaystyle X}$, the following are equivalent:^[2]

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.^[8]


Hausdorff spaces are T₁, meaning that each singleton is a closed set. Similarly, preregular spaces are R₀. Every Hausdorff space is a Sober space although the converse is in general not true.


Another property of Hausdorff spaces is that each compact set is a closed set. For non-Hausdorff spaces, it can be that each compact set is a closed set (for example, the cocountable topology on an uncountable set) or not (for example, the cofinite topology on an infinite set and the 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,^[9] 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.^[10][11] Compact preregular spaces are normal,^[12] 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 ${\displaystyle f\colon X\to Y}$ be a continuous function and suppose ${\displaystyle Y}$ is Hausdorff. Then the graph of ${\displaystyle f}$, ${\displaystyle \{(x,f(x))\mid x\in X\}}$, is a closed subset of ${\displaystyle X\times Y}$.


Let ${\displaystyle f\colon X\to Y}$ be a function and let ${\displaystyle \ker(f)\triangleq \{(x,x')\mid f(x)=f(x')\}}$ be its kernel regarded as a subspace of ${\displaystyle X\times X}$.


If ${\displaystyle f,g\colon X\to Y}$ are continuous maps and ${\displaystyle Y}$ is Hausdorff then the equalizer ${\displaystyle {\mbox{eq}}(f,g)=\{x\mid f(x)=g(x)\}}$ is a closed set in ${\displaystyle X}$. It follows that if ${\displaystyle Y}$ is Hausdorff and ${\displaystyle f}$ and ${\displaystyle g}$ agree on a dense subset of ${\displaystyle X}$ then ${\displaystyle f=g}$. In other words, continuous functions into Hausdorff spaces are determined by their values on dense subsets.


Let ${\displaystyle f\colon X\to Y}$ be a closed surjection such that ${\displaystyle f^{-1}(y)}$ is compact for all ${\displaystyle y\in Y}$. Then if ${\displaystyle X}$ is Hausdorff so is ${\displaystyle Y}$.


Let ${\displaystyle f\colon X\to Y}$ be a quotient map with ${\displaystyle X}$ a compact Hausdorff space. Then the following are equivalent:

Preregularity versus regularity[edit]

All regular spaces are preregular, as are all Hausdorff spaces. There are many results for topological spaces that hold for both regular and Hausdorff spaces. Most of the time, these results hold for all preregular spaces; they were listed for regular and Hausdorff spaces separately because the idea of preregular spaces came later. On the other hand, those results that are truly about regularity generally do not also apply to nonregular Hausdorff spaces.


There are many situations where another condition of topological spaces (such as paracompactness or local compactness) will imply regularity if preregularity is satisfied. Such conditions often come in two versions: a regular version and a Hausdorff version. Although Hausdorff spaces are not, in general, regular, a Hausdorff space that is also (say) locally compact will be regular, because any Hausdorff space is preregular. Thus from a certain point of view, it is really preregularity, rather than regularity, that matters in these situations. However, definitions are usually still phrased in terms of regularity, since this condition is better known than preregularity.


See History of the separation axioms for more on this issue.