Sierpiński room

The Sierpiński space is a topological space , consisting of two points, in which exactly a lot is open and not closed at the same time. It is the smallest room with a non- discrete and non- trivial topology.

definition

The set of points on which the Sierpiński space is based is ; its open sets are and . ${\displaystyle \mathbb {S} }$ ${\displaystyle \{\bot ,\top \}}$ ${\displaystyle \emptyset ,\{\top \}}$ ${\displaystyle \{\bot ,\top \}}$

Relationship to other topological spaces

If an arbitrary set is a two-element set, then every function corresponds to a subset , and vice versa. ${\displaystyle M}$ ${\displaystyle 2:=\{0,1\}}$ ${\displaystyle \chi _{A}\colon M\to 2}$ ${\displaystyle A\subseteq M}$

A role that is too analogous is assumed in the case of continuous functions and open subsets. Let be any topological space. For a continuous function , according to the definition for continuous functions, it holds that the archetypes of open sets are open. and . Provides an interesting result . This is namely an open subset of and is uniquely determined by the continuous . ${\displaystyle 2}$ ${\displaystyle \mathbb {S} }$ ${\displaystyle M}$ ${\displaystyle \chi \colon M\to \mathbb {S} }$ ${\displaystyle \chi ^{-1}(\{\bot ,\top \})=M}$ ${\displaystyle \chi ^{-1}(\emptyset )=\emptyset }$ ${\displaystyle \chi ^{-1}(\{\top \})}$ ${\displaystyle M}$ ${\displaystyle \chi }$

The sierpiński space is cogenerator the category of Kolmogorov rooms : are continuous maps between two Kolmogorov-rooms and with , then there exists a continuous mapping , so : Let this with , so at least through an open area of separated, or vice versa (as is a Kolmogorov room). Then delivers the desired . In fact, the cogenerators of the Kolmogorov space category are all Kolmogorov spaces that contain a subspace that is homeomorphic to . ${\displaystyle f,g\colon A\to B}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle f\neq g}$ ${\displaystyle d\colon B\to \mathbb {S} }$ ${\displaystyle df\neq dg}$ ${\displaystyle x\in A}$ ${\displaystyle f(x)\neq g(x)}$ ${\displaystyle f(x)}$ ${\displaystyle U}$ ${\displaystyle g(x)}$ ${\displaystyle B}$ ${\displaystyle \chi _{U}}$ ${\displaystyle d}$ ${\displaystyle \mathbb {S} }$

Individual evidence

1. ↑ Dieter Pumplün : Elements of the category theory . Spectrum - Akademischer Verlag, Heidelberg et al. 1999, ISBN 3-86025-676-9 , p. 80 .

literature

• Lynn Arthur Steen , J. Arthur Seebach: Counterexamples in Topology . Dover Publications, New York NY 1995, ISBN 0-486-68735-X ( MR 507446 ).