# 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. ${\ displaystyle \ {x_ {0} \} \ hookrightarrow X}$

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 . ${\ displaystyle \ {\ star \} \ downarrow \ operatorname {Top}}$${\ displaystyle X \ vee Y}$

## Homotopy classes of dotted maps

Two dotted figures

${\ displaystyle f, g \ colon (X, x_ {0}) \ to (Y, y_ {0})}$

hot homotopic if there is a continuous map with ${\ displaystyle H \ colon X \ times \ left [0,1 \ right] \ to Y}$

${\ displaystyle H (x, 0) = f (x), H (x, 1) = g (x) \ \ forall x \ in X}$
${\ displaystyle H (x_ {0}, t) = y_ {0} \ \ forall t \ in \ left [0,1 \ right]}$

gives. The set of homotopy classes of dotted maps is denoted by. ${\ displaystyle \ left [X, Y \ right]}$

## 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.