Hausdorff space

${\displaystyle U}$

${\displaystyle x}$

${\displaystyle V}$

${\displaystyle y}$

${\displaystyle U}$

${\displaystyle V}$

${\displaystyle (U\cap V=\varnothing )}$

${\displaystyle X}$

${\displaystyle X}$

A related, but weaker, notion is that of a preregular space. ${\displaystyle X}$ is a preregular space if any two

The relationship between these two conditions is as follows. A topological space is Hausdorff

Equivalences

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

• ${\displaystyle X}$ is a Hausdorff space.
• Limits of
  nets
  in ${\displaystyle X}$ are unique.^[3]
• Limits of
  filters
  on ${\displaystyle X}$ are unique.^[3]
• Any
  singleton set
  ${\displaystyle \{x\}\subset X}$ is equal to the intersection of all closed neighbourhoods of ${\displaystyle x}$.^[4] (A closed neighbourhood of ${\displaystyle x}$ is a closed set that contains an open set containing ${\displaystyle x}$.)
• The diagonal ${\displaystyle \Delta =\{(x,x)\mid x\in X\}}$ is
  product space
  ${\displaystyle X\times X}$.
• Any injection from the discrete space with two points to ${\displaystyle X}$ has the lifting property with respect to the map from the finite topological space with two open points and one closed point to a single point.

Examples of Hausdorff and non-Hausdorff spaces

Almost all spaces encountered in

In contrast, non-preregular spaces are encountered much more frequently in abstract algebra and algebraic geometry, in particular as the Zariski topology on an algebraic variety or the spectrum of a ring. They also arise in the model theory of intuitionistic logic: every complete Heyting algebra is the algebra of open sets of some topological space, but this space need not be preregular, much less Hausdorff, and in fact usually is neither. The related concept of Scott domain also consists of non-preregular spaces.

While the existence of unique limits for convergent nets and filters implies that a space is Hausdorff, there are non-Hausdorff T₁ spaces in which every convergent sequence has a unique limit.

Properties

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

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

${\displaystyle X\times X}$

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

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

${\displaystyle X}$

${\displaystyle f=g}$

Let ${\displaystyle f\colon X\to Y}$ be a

${\displaystyle f^{-1}(y)}$

${\displaystyle y\in Y}$

${\displaystyle X}$

${\displaystyle Y}$

Let ${\displaystyle f\colon X\to Y}$ be a

${\displaystyle X}$

• ${\displaystyle Y}$ is Hausdorff.
• ${\displaystyle f}$ is a
  closed map
  .
• ${\displaystyle \ker(f)}$ is a closed set.

Preregularity versus regularity

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

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

Variants

The terms "Hausdorff", "separated", and "preregular" can also be applied to such variants on topological spaces as uniform spaces, Cauchy spaces, and convergence spaces. The characteristic that unites the concept in all of these examples is that limits of nets and filters (when they exist) are unique (for separated spaces) or unique up to topological indistinguishability (for preregular spaces).

As it turns out, uniform spaces, and more generally Cauchy spaces, are always preregular, so the Hausdorff condition in these cases reduces to the T₀ condition. These are also the spaces in which

Algebra of functions

The algebra of continuous (real or complex) functions on a compact Hausdorff space is a commutative C*-algebra, and conversely by the Banach–Stone theorem one can recover the topology of the space from the algebraic properties of its algebra of continuous functions. This leads to noncommutative geometry, where one considers noncommutative C*-algebras as representing algebras of functions on a noncommutative space.

Academic humour

• Hausdorff condition is illustrated by the pun that in Hausdorff spaces any two points can be "housed off" from each other by
  open sets.^[13]
• In the Mathematics Institute of the
  University of Bonn, in which Felix Hausdorff researched and lectured, there is a certain room designated the Hausdorff-Raum. This is a pun
  , as Raum means both room and space in German.

• Fixed-point space – Topological space such that every endomorphism has a fixed point, a Hausdorff space X such that every continuous function f : X → X has a fixed point.
• Locally Hausdorff space
• Non-Hausdorff manifold – generalization of manifoldsPages displaying wikidata descriptions as a fallback
• Quasitopological space – a set X equipped with a function that associates to every compact Hausdorff space K a collection of maps K→C satisfying certain natural conditionsPages displaying wikidata descriptions as a fallback
• Separation axiom – Axioms in topology defining notions of "separation"
• Weak Hausdorff space – concept in algebraic topologyPages displaying wikidata descriptions as a fallback

Notes