category of extended metric spaces

 analysis
Notation Met∞\Met_{\infty} Objects extended metric spaces, i.e. metric spaces where the metric is allowed to assume the value ∞\infty Morphisms non-expansive maps ff (also called short maps or 11-Lipschitz maps), meaning d(f(x),f(y))≤d(x,y)d(f(x),f(y)) \leq d(x,y) for all x,yx,y Related Met\Met, Metc\Met_c External nLab link
The fact that we allow ∞\infty means that universal constructions work much better when compared to Met\Met.

Satisfied properties

Assigned properties

Deduced properties

Unsatisfied properties

Assigned properties

Deduced properties*

*This also uses the deduced satisfied properties.

Unknown properties

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Special objects

  • terminal object: singleton space
  • initial object: empty metric space
  • products: direct products with the metric d(x,y)=sup⁡idi(xi,yi)d(x,y) = \sup_i d_i(x_i,y_i)
  • coproducts: disjoint union with the metric that extends the given ones and gives points in different spaces the distance ∞\infty

Special morphisms