category of metric spaces with continuous maps

This category is equivalent to the subcategory of Top\mathbf{Top} (or Haus\mathbf{Haus}) that consists of metrizable topological spaces. Hence, the metrics only play a secondary role here.

Satisfied Properties

Properties from the database

Deduced properties

Unsatisfied Properties

Properties from the database

Deduced properties*

*This also uses the deduced satisfied properties.

Unknown properties

There are 4 properties for which the database doesn't have an answer if they are satisfied or not. Please help to contribute the data!

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(x,y)/2id(x,y) = \sum_i d_i(x,y) / 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: embeddings of closed subspaces
  • regular epimorphisms: