Totally separated room

Totally separated rooms are examined in the mathematical sub-area of topology . In a disconnected topological space there is at least one non-empty and open-closed set different from the overall space ; in totally separated spaces there are very many of them.

U is open-ended, contains x but not y.

definition

A topological space is completely separated if for any two distinct points of an open-closed set is with and . ${\ displaystyle X}$ ${\ displaystyle x \ not = y}$ ${\ displaystyle X}$ ${\ displaystyle U \ subset X}$ ${\ displaystyle x \ in U}$ ${\ displaystyle y \ notin U}$

Note that the definition is symmetrical with respect to and , because with is also open-ended. ${\ displaystyle x}$ ${\ displaystyle y}$ ${\ displaystyle U}$ ${\ displaystyle X \ setminus U}$

Examples

Discrete rooms are totally separated. In general, one has the following relationships for Hausdorff spaces , which provide further examples:

Discrete, extremal, totally disconnected, totally disconnected .

{\ displaystyle \ Rightarrow}

{\ displaystyle \ Rightarrow}

{\ displaystyle \ Rightarrow}

The dotted Knaster-Kuratowski fan is totally incoherent but not totally separated.

Zero-dimensional T ₀ spaces are totally separated. Zero-dimensional spaces do not fit into the above series, since there are extremely disconnected Hausdorff spaces that are not zero-dimensional. At the same time, this shows that even in the case of Hausdorff spaces, their total separation does not result in their zero-dimensionality, as the following example shows:

On the set of irrational numbers is a lot of open exactly when it all one there, so from and follows that applies. This defines a topology that is totally separated but not zero-dimensional. ${\ displaystyle \ mathbb {R} {\ setminus} \ mathbb {Q}}$ ${\ displaystyle U \ subseteq \ mathbb {R} {\ setminus} \ mathbb {Q}}$ ${\ displaystyle x \ in U}$ ${\ displaystyle \ delta> 0}$ ${\ displaystyle y \ in {\ sqrt {2}} \ cdot \ mathbb {Q}}$ ${\ displaystyle | xy | <\ delta}$ ${\ displaystyle y \ in U}$ ${\ displaystyle \ mathbb {R} {\ setminus} \ mathbb {Q}}$

properties

Sub-rooms of totally separated rooms are totally separated again.
Products of totally separated rooms are totally separated again.
Every totally separated room is Hausdorff-like, because the defining property provides open spaces separating two points.
A locally compact Hausdorff space is totally separated if and only if it is totally disconnected.

Individual evidence

↑ René Bartsch: Allgemeine Topologie , Walter De Gruyter (2015), ISBN 978-3-11-040617-7 , sentence 6.4.4
↑ Lynn Arthur Steen, J. Arthur Seebach: Counterexamples in Topology. Springer-Verlag, 1978, ISBN 3-540-90312-7 , example 113
^ Peter T. Johnstone: Stone Spaces , Cambridge University Press 1982, ISBN 0-521-33779-8 , II.4.2 Exercise (ii)
↑ Michel Coornaert: Topological Dimension and Dynamical Systems , Springer-Verlag 2015, ISBN 978-3-319-19793-7 , sentence 2.6.3
↑ René Bartsch: General Topology , Walter De Gruyter (2015), ISBN 978-3-11-040617-7 , Corollary 6.4.8

[1] René Bartsch: Allgemeine Topologie , Walter De Gruyter (2015), ISBN 978-3-11-040617-7 , sentence 6.4.4

[2] Lynn Arthur Steen, J. Arthur Seebach: Counterexamples in Topology. Springer-Verlag, 1978, ISBN 3-540-90312-7 , example 113

[3] Peter T. Johnstone: Stone Spaces , Cambridge University Press 1982, ISBN 0-521-33779-8 , II.4.2 Exercise (ii)

[4] Michel Coornaert: Topological Dimension and Dynamical Systems , Springer-Verlag 2015, ISBN 978-3-319-19793-7 , sentence 2.6.3

[5] René Bartsch: General Topology , Walter De Gruyter (2015), ISBN 978-3-11-040617-7 , Corollary 6.4.8