In topology and related branches of mathematics, Hausdorff space, separation space, or TtwoSpace is a topological space in which all points are "separated by neighborhood".Among many separation axioms that can be imposed on topological space, "Hausdorff condition" is the most frequently used and discussed.It implies the uniqueness of the limit of sequence, net and filter.Hausdorff is named after Felix Hausdorff, one of the founders of topology.The initial definition of Hausdorff topological space includes Hausdorff conditions asaxiom。
Suppose X is a topological space.Let x and y be points in X.If there is a neighborhood U of x and a neighborhood V of y so that U and V are disjoint (U ∨ V=∅), we say that x and y can be "separated by the neighborhood".X is a Hausdorff space if any two different points of X can be separated by the neighborhood.This is the Hausdorff space, also calledT2 spaceAnd separation of space.
X is a preregular space, if any two topologically distinguishable points can be separated by the neighborhood.Preregular spaces are also called R1 spaces.
The relationship between these conditions is as follows.A topological space is a Hausdorff space if and only if it is a preregular space andKolmogorov spaceThat is to say, unique points are topologically distinguishable.A topological space is a preregular space if and only if its Kolmogorov quotient space is a Hausdorff space.
equivalence
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For topological space X, the following statements are equivalent:
X is the Hausdorff space.
X is a closed set of product spaces.
The limit in X is unique (i.esequence、networkandFilterConverges to at most one point).
All contained in XSingle element setIs equal to the intersection of all its closed neighbors.
The diagonal Δ={(x, x) | x ∈ X} is a closed set as a subset of the product space X × X.
Examples and counterexamples
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Almost all spaces encountered in mathematical analysis are Hausdorff spaces;The most important real number is Hausdorff space.More generally, all metric spaces are Hausdorff spaces.In fact, many spaces used in the analysis, such as topological groups and topological manifolds, explicitly state Hausdorff conditions in their definitions.
The simplest is ToneAn example of the topology of a space other than T2 space is cofinite space.
Pseudometric spaces are typically not Hausdorff spaces, but they are preregular, and they are usually only used to construct Hausdorff gauge spaces in analysis.In fact, when analysts deal with non Hausdorff space, it should at least be preregular. They simply replace it with its Kolmogorov quotient space of Hausdorff space.
On the contrary, non preregular spaces are more often seen in abstract algebra and algebraic geometry, especially inAlgebraic varietyOr commutative ringZarisky topology。They also appear in the model theory of intuitive logic: all completeHeyting algebraAll are algebras of open sets of a topological space, but this space does not need to be preregular, and Hausdorff space is rare.
Local compactness
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Let X be a Hausdorff space, then the following conditions are equivalent:
Two nonempty topological spaces are Hausdorff spaces if and only ifProduct spaceIs a Hausdorff space if and only if itdisjoint union It's a Hausdorff space.[2]
X'sQuotient spaceIt does not have to be a Hausdorff space.In fact, all topological spaces can be implemented as quotients of some Hausdorff space.
X is ToneSpace, which means that all single element sets are closed sets.Similarly, preregular spaces are R0 spaces.
X'sCompact setalwaysClosed set。This may fail for non Hausdorff space (for example, T with its failureoneSpace).
If K is a compact set of X and F is a closed set of X, then F ∨ K is a compact set.[4]
The definition of Hausdorff space claims that points can be separated by neighbors.It implies something stronger in representation: all pairs of disjoint compact sets in a Hausdorff space can be separated from neighbors.This is an example of the general rule that compact sets often behave as points.
Together with preregularity, compactness conditions often imply stronger separation axioms.For example, any locally compact preregular space is a completely regular space.Compact preregular spaces are normal spaces, meaning that they satisfyUlysson lemmaandTitzer expansion theoremAnd there is a unit partition subject to local finite open cover.The Hausdorff version of these statements is that all locally compact Hausdorff spaces areTychonoff space All compact Hausdorff spaces are normal Hausdorff spaces.
The following results are about the technical properties of mappings (continuous functions and others) from or to Hausdorff spaces.
Let f: X → Y be a continuous function and let Y be a Hausdorff space.Then the image of f is a closed subset of X × Y.
Let f: X → Y be a function and its kernel as a subspace of X × X.
If f is a continuous function and Y is a Hausdorff space, then ker (f) is a closed set.
If f is an open surjection and ker (f) is a closed set, then Y Hausdorff space.
If f is a continuous open epimorphism (that is, an open quotient mapping), then Y is a Hausdorff space if and only if ker (f) is a closed set.
If f, g: X → Y is a continuous mapping and Y Hausdorff space, then the equalizer is a closed set in X.It can be concluded that if Y is a Hausdorff space and f and g are consistent with the dense subset of X, then f=g.In other words, continuous functions into a Hausdorff space determine their values on a dense subset.
Let f: X → Y be a closed surjection such that f − 1 (y) is compact for all y ∈ Y.If X is a Hausdorff space, so is Y.
Let f: X → Y be a quotient map with X being a compact Hausdorff space.Then the following are equivalent
Y is the Hausdorff space
F is a closed mapping
Ker (f) is a closed set
Preregularity and regularity
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All regular spaces are preregular spaces, and they are also Hausdorff spaces.There are many results of topological space for both regular space and Hausdorff space.Most of the time, these results are also true for all preregular spaces;They list regular space and Hausdorff space separately, because the concept of preregular space comes later.On the other hand, these results which are true for regularity are generally not applicable to irregular Hausdorff spaces.
In many cases, other conditions of topological space (such as paracompactness or local compactness) also imply regularity if it satisfies preregularity.There are often two versions of this condition: the regular version and the Hausdorff version.Although Hausdorff spaces are generally not regular, locally compact Hausdorff spaces are regular, because any Hausdorff space is preregular.Therefore, from a specific point of view, it is actually preregular rather than regular in these cases.However, the definition is still worded on the basis of regularity, because these conditions are more well known than preregularity.
See the history of the separation axiom for more details.
variant
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The terms "Hausdorff", "separation" and "preregularization" can also be used in topological space variants such as uniform space, Cauchy space and convergence space.In all these examples, the unified conceptual feature is that the limit of a net or filter (when they exist) is unique (for separated spaces) or unique in the sense of topological isomorphism (for preregular spaces).
This shows that uniform spaces and more general Cauchy spaces are always preregular, so in these cases the Hausdorff condition is reduced to T0 condition.There is also a meaningful space for completeness, and Hausdorff is the natural partner of completeness in these cases.In particular, a space is complete if and only if all Cauchy nets have at least one limit, and a space is Hausdorff's, if and only if all Cauchy nets have at most one limit (because only Cauchy nets can have a limit first).
Weak hausdorff space
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If any mapping from compact Hausdorff space K to X
, g (K) is a closed subset of X, then X isWeak hausdorff space。If X is a weak Hausdorff space, then g (K) is a compact Hausdorff subspace of X.[1]
quote
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Munkres, J. R., 2000, Topology, 2nd edition, Upper Saddle River, NJ: Prentice Hall. ISBN 0-131-81629-2
Zhao Wenmin, Introduction to Topology, Jiuzhang Publishing House, ISBN 957-603-018-8
Arkhangelskii, A.V., L.S.Pontryagin, General Topology I, (1990) Springer-Verlag, Berlin. ISBN 3-540-18178-4
Bourbaki; Elements of Mathematics: General Topology, Addison-Wesley (1966).
