## Continuous Maps Between Metric Spaces

The map between metric spaces is continuous if it satisfies the following properties, which are equivalent:

1. For every open set in is open in This is the general topological definition of continuity. This means than if then an open set containing is sent to an open set containing under 2. If is a sequence in that converges to then the sequence converges to 3. For every and every there exists such that for all we have In general every space may have more than one possible metric. For fixed may depend on the metric chosen.

Moreover, is continuous if and only if it is continuous on every compact subset of The image of every closed set is closed, the image of every compact set under a continuous function is compact, and the image of every connected set under a continuous function is connected. It is not the case however that the image of every open set is open. If is the constant map from to with the Euclidean metric, then is a single point hence is closed in  