category of metric spaces with continuous maps

notation: $\mathbf{Met}_c$
objects: metric spaces
morphisms: continuous maps
Related categories: $\mathbf{Haus}$ , $\mathbf{Met}$ , $\mathbf{Met}_{\infty}$ , $\mathbf{Top}$

This category is equivalent to the subcategory of $\mathbf{Top}$ (or $\mathbf{Haus}$ ) that consists of metrizable topological spaces.

Properties from the database

Deduced properties

Properties from the database

Deduced properties*

*This also uses the deduced satisfied properties.

There are 3 properties for which the database doesn't have an answer if they are satisfied or not. Please help to contribute the data!

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