category of metric spaces with continuous maps

 analysis
Notation Metc\Met_c Objects metric spaces Morphisms continuous maps Related Haus\Haus, Met\Met, Met∞\Met_{\infty}, Top\Top, Unif\Unif, Euclid⊔\Euclidsqcup External nLab link
This category is equivalent to the subcategory of Top\Top (or Haus\Haus) that consists of metrizable topological spaces. Hence, the metrics only play a secondary role here.

Satisfied properties

Assigned properties

Deduced properties

Unsatisfied properties

Assigned properties

Deduced properties*

*This also uses the deduced satisfied properties.

Unknown properties

There is 1 property for which the database has no information on whether it is satisfied. Please help us fill in the gaps by contributing to this project.

Special objects

  • terminal object: singleton space
  • initial object: empty metric space
  • products: [countable case] In the finite case, take direct products with the metric d(x,y)=sup⁡idi(xi,yi)d(x,y) = \sup_i d_i(x_i,y_i), but other metrics such as d(x,y)=∑idi(xi,yi)d(x,y) = \sum_i d_i(x_i,y_i) also work. In the countable case, one can assume di≤1d_i \leq 1 and then define d(x,y)=∑idi(xi,yi)/2id(x,y) = \sum_i d_i(x_i,y_i) / 2^i.
  • coproducts: Given metric spaces (Xi,di)(X_i,d_i) with di≤1d_i \leq 1 w.l.o.g., we endow the disjoint union ∐iXi\coprod_i X_i with the metric dd that extends the metrics did_i and satisfies d(x,y)=1d(x,y) = 1 when x,yx,y are in different XiX_i.

Special morphisms

  • isomorphisms: homeomorphisms
  • monomorphisms: injective continuous maps
  • epimorphisms: continuous maps with dense image
  • regular monomorphisms: closed embeddings
  • regular epimorphisms: A continuous map f:X→Yf : X \to Y of metric spaces is a regular epimorphism if and only if ff is surjective and YY carries the "final metric topology" with respect to ff; by this we mean that, for every metric space ZZ (not necessarily every topological space) and every set map g:Y→Zg : Y \to Z such that g∘fg \circ f is continuous, gg is continuous. Furthermore, it is sufficient to demand this for Z=RZ = \IR.