Normal room

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Note : There is no uniform understanding of the terms normal room and T 4 room in the standard literature ; rather, there is inconsistency. In this article the view applies that a T 4 room is a normal Hausdorff room , while a normal room does not necessarily have to be Hausdorff-like .

Graphic representation of a normal room

A normal space is a topological space in which any two disjoint closed sets have disjoint neighborhoods . Less: Completed quantities E , F are environments U , V separated .

This property is, for example, the basis of Urysohn's lemma or Tietze's continuation theorem . The term goes back to Heinrich Tietze in 1923, its full scope was recognized by Urysohn in his work on the continuation of functions.

Normality is not necessarily passed on to all subspaces.

motivation

A common method for examining an object in a mathematical category is to examine the set of structure-maintaining functions in particularly well-understood representatives of the category. In many cases one can also gain knowledge about the object to be examined in this way. In linear algebra , for example, one examines the set of linear mappings from any vector space into the basic field and denotes this as the dual space . In topology, the topological spaces and are available as model spaces .

In terms of continuity, however, this approach can only make sense if additional conditions are placed on the space to be examined. In a space with the trivial topology, for example, every continuous complex-valued function is already constant (this even applies to every continuous function whose target set is a Kolmogoroff space ).

If one wants to understand a topological space by examining the continuous functions of it in one of the model spaces, the set of these functions should at least be point-separating . This leads to the definition of a complete Hausdorff space . This is precisely defined by the existence of a sufficient set of continuous functions.

It would of course be desirable to have an elementary topological criterion that ensures this existence. Hausdorff rooms that are normal or locally compact are ideal here. Most of the topological spaces examined in mathematics fall into at least one of the two categories. The lemma Urysohn provides for these two classes of rooms (among other things) to ensure that they are complete Hausdorff spaces.

In fact, the more general continuation of Tietze shows that in such spaces continuous functions in one of the model spaces, which are only defined on a closed (for normal spaces) or compact (for locally compact spaces) subset, become continuous functions of the entire space in the Let the model room continue. In the second case, the continuation can be chosen so that it still has a compact carrier .

Formal definition of the normal room and the T 4 room (normal Hausdorff room)

It should be noted that the definition in the literature is inconsistent, here the Hausdorffsch property is not required for a normal room, but it is for a T 4 room.

Be a topological space. is normal , if for any two closed subsets , with environments , and are of E and F with .

A normal room that also fulfills the separation property T 2 , that is to say is a normal Hausdorff room , is referred to as a T 4 room.

Many authors use the terms differently: They automatically assume Hausdorffsch for a normal room (i.e. T 2 room) and understand T 4 rooms to be the room class described in this article under "normal", so the requirement that T 4 -Rooms are Hausdorff-like. Most of the normal rooms occurring in the applications are T 2 rooms.

Examples

properties

Inheritance properties

Continuation of continuous functions

A topological space is exactly then a normal room when each on a closed subset continuous , real-valued function continuous to a whole on the space, real-valued function continues to be.

Lemma from Urysohn

A topological space is a normal space if and only if there is a continuous function with and for every two disjoint, closed sets .

Closed environments

A simple reformulation of the definitions yields:

A topological space is normal if and only if there is an open set for every neighborhood of a closed set , for which the following applies:

This means that for each closed set, the closed environments form an environment basis .

Decomposition of the one

A normal space allows a decomposition of the one for every locally finite open cover.

Overlaps

A T 1 -space is normal if and only if every open, locally finite cover has a shrinkage , that is, there is an open cover with for all .

Specializations

The concept of normal space can be sharpened in several ways:

  • A normal space is called completely normal if there are two sets with disjoint open sets and with and . So there is a stronger separation property here. In such rooms, all sub-rooms, not just the closed ones, are normal. The Tikhonov plank is a non-normal subspace of a compact , the latter is therefore normal but not completely normal.
  • A normal space is called perfectly normal if there is a continuous function with and for every two disjoint closed sets . A stronger version of Urysohn's lemma applies in such spaces. The one-point compactification of the Tichonow plank is not perfectly normal, since the infinitely distant point is not a set and therefore cannot be a zero formation of a continuous, real-valued function.
  • A normal space is called totally normal if there is an open cover for every open set such that
    • Each is a -set , that is, a countable union of closed sets.
    • is locally finite on , i. H. for everyone there is an environment that has a non-empty section with only a finite number of them.
Such spaces play a role in dimensional theory . Perfectly normal rooms are totally normal.

literature

  • Boto von Querenburg : Set theoretical topology (=  Springer textbook ). 3rd revised and expanded edition. Springer Verlag, Berlin (among others) 2001, ISBN 3-540-67790-9 .
  • Egbert Harzheim , Helmut Ratschek: Introduction to General Topology (=  Mathematics. Introductions to the subject matter and results of its sub-areas and related sciences ). Scientific Book Society, Darmstadt 1978, ISBN 3-534-06355-4 . MR0380697
  • Horst Schubert : Topology (=  mathematical guidelines ). 4th edition. BG Teubner Verlag, Stuttgart 1975, ISBN 3-519-12200-6 ( MR0423277 ).
  • Stephen Willard: General Topology . Addison-Wesley, Reading MA et al. a. 1970 ( MR0264581 ).

Individual evidence

  1. Stephen Willard: General Topology . Addison-Wesley, Reading MA et al. a. 1970, p. 99 ( MR0264581 ).
  2. Schubert (p. 77), for example, calls a normal room one that is referred to in this article as T 4 room .
  3. ^ Heinrich Tietze : Contributions to the general topology I. Axioms for different versions of the concept of environment. In: Mathematical Annals. 88, 1923, ISSN  0025-5831 , pp. 290–312, ursinus.edu  ( page no longer available , search in web archivesInfo: The link was automatically marked as defective. Please check the link according to the instructions and then remove this notice. (PDF; 1.23 MB).@1@ 2Template: dead link / webpages.ursinus.edu  
  4. ^ N. Bourbaki : Éléments d'histoire des mathématiques. Springer, Berlin a. a. 2007, ISBN 978-3-540-33938-0 , p. 205.
  5. René Bartsch: General Topology. Walter de Gruyter GmbH & Co KG, 2015, ISBN 978-3-11-040618-4 , p. 124, Lemma 4.4.13.
  6. ^ Karl Peter Grotemeyer : Topologie , Bibliographisches Institut Mannheim (1969), ISBN 3-411-00836-9 , sentence 43.