nLab
continuous space

Analogously to a smooth space, a continuous space is any notion of space between which the morphisms are continuous maps. Examples include:

• convergence spaces,
• topological spaces,
• locales.

Contrast this with, for example:

• uniform spaces (morphisms are uniformly continuous maps),
• smooth spaces (morphisms are smooth maps),
• (weak) homotopy types (objects are still (nice) topological spaces, but now morphisms are homotopy classes of continuous maps).