Induced topology from metric is functorial

Statement
The association that sends a metric space to its induced topological space (where the open balls form a basis) gives a functor from the category of metric spaces with continuous maps to the category of topological spaces with continuous maps. In other words, a continuous map of metric spaces gives rise to a continuous map of the corresponding topological spaces.

Moreover, the functor is full: a map is continuous as a map of metric spaces if and only if it is continuous as a map of topological spaces.