Dotted topological space

from Wikipedia, the free encyclopedia

A pointed space is a pair ( X , x 0 ), consisting of a topological space X and a point x 0 in X ( ground point , basic point , excellent spot ). A dotted (continuous) map ( X , x 0 ) → ( Y , y 0 ) is a continuous map X  →  Y , which maps x 0 to y 0 .

Often the base point is simply designated with an asterisk.

If the inclusion is a cofiber , one speaks of a well-dotted space.

A topological space is called homogeneous if two dotted topological spaces are isomorphic on it.

Categorical properties

The category of dotted topological spaces is isomorphic to the comma category . It has zero objects (those spaces which only consist of one point). Products are the usual products of topological spaces, coproducts are one-point unions , i.e. disjoint unions in which the respective marked points are identified with each other, written .

Homotopy classes of dotted maps

Two dotted figures

hot homotopic if there is a continuous map with

gives. The set of homotopy classes of dotted maps is denoted by.

Individual evidence

  1. Jon P. May: A Concise Course in Algebraic Topology. University of Chicago Press, Chicago IL et al. 1999, ISBN 0-226-51183-9 , section 8.3.