Continuous map of metric spaces

Definition

In terms of the metric

Suppose $(X, d_{X})$ and $(Y, d_{Y})$ are Metric space (?)s. In other words, $X$ and $Y$ are sets, and $d_{X}$ and $d_{Y}$ are metrics on these sets. A function $f : X \to Y$ is termed a continuous map from $X$ to $Y$ if it satisfies the following:

$\forall ε > 0, a \in X, \exists δ > 0 : d_{X} (a, b) < δ ⟹ d_{Y} (f (a), f (b)) < ε$ .

In terms of the induced topology

Suppose $(X, d_{X})$ and $(Y, d_{Y})$ are metric spaces. A function $f : X \to Y$ is termed a continuous map if $f$ is a continuous map from $X$ to $Y$ with the induced topologies from their respective metrics.

The notion of continuous map of metric spaces gives rise to the notion of the category of metric spaces with continuous maps.