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.