Hausdorff Space Definitions Alternative Characterisations

X is a preregular space, or R₁ space, iff, given any topologically distinguishable points x and y, x and y can be separated by neighbourhoods.

2 Alternative characterisations

In fact, a topological space is Hausdorff if and only if every net (or filter) has at most one limit. Similarly, a space is preregular iff all of the limits of a given net (or filter) are topologically indistinguishable.

A topological space X is Hausdorff if and only if the diagonal {(x,x) : x in X} is a closed setIn topology and related branches of mathematics, a set is called closed if its complement is open. This implies that a closed set contains its own boundary. Intuitively, if you are outside the set, and you "wiggle" a little bit, you will still be outside in X × X, the Cartesian productIn mathematics, the Cartesian product (or direct product X × Y of two sets X and Y is the set of all ordered pairs whose first component is a member of X and whose second component is a member of Y''. This concept is named after Rene Descartes. X × Y { x of X with itself.

A topological space is Hausdorff if and only if it is both preregular and T₀In topology and related branches of mathematics, the T spaces or Kolmogorov spaces form a broad class of well behaved topological spaces. The T condition is one of the separation axioms. Topological distinguishability To define T spaces, we first define t. Conversely, a topological space is preregular if and only if its Kolmogorov quotient is Hausdorff.

3 Nomenclature

4 Examples and nonexamples

Almost all spaces encountered in analysisAnalysis is that branch of mathematics which deals with the real numbers and complex numbers and their functions. It has its beginnings in the rigorous formulation of calculus and studies concepts such as continuity, integration and differentiability in g are Hausdorff; most importantly, the real numberIn mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. The term "real number" is a retronym coined in response to " imaginary number". Real numbers mays are a Hausdorff space. More generally, all metric spaces are Hausdorff. In fact, many spaces of use in analysis, such as topological groups and topological manifolds, have the Hausdorff condition explicitly stated in their definitions.

Indeed, when analysts run across a non-Hausdorff space, it is still probably at least preregular, and then they simply replace it with its Kolmogorov quotient, which is Hausdorff.

5 Properties

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

Subspaces and products of Hausdorff spaces are Hausdorff, but quotient spaces of Hausdorff spaces need not be Hausdorff.

In fact, a quotient space of a Hausdorff space X is itself Hausdorff if and only if the kernel of the quotient map is closed as a subset of the Cartesian product X × X.

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.

6 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 don't 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 aren't generally 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 more well known than preregularity.

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

7 Variants

The terms "Hausdorff", "separated", and "preregular" can also be applied to such variants on topological spaces as uniform spaces, Cauchy spaces, and convergence space s. 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 completeness makes sense, and Hausdorffness is a natural companion to completeness in these cases. Specifically, a space is complete iff every Cauchy net has at least one limit, while a space is Hausdorff iff every Cauchy net has at most one limit (since only Cauchy nets can have limits in the first place).

8 Joke

There is a mathematicians' joke that serves as a reminder of the meaning of this term: In a Hausdorff space, points can be "housed off" from one another. Atiyah used to draw house-shaped sets, on the blackboard. (In an old-fashioned British accent, off could be orf, phonetically, which all helps.)

General topology Topology

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1 Definitions