Dotted topological space
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
- ↑ 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.