Continuous Maps Between Metric Spaces
The mapbetween metric spaces is continuous if it satisfies the following properties, which are equivalent:
- 1. For every open setinis open in
- This is the general topological definition of continuity.
- This means than ifthen an open set containingis sent to an open set containingunder
- 2. Ifis a sequence inthat converges tothen the sequenceconverges to
- 3. For everyand everythere existssuch that for allwe haveIn general every space may have more than one possible metric. For fixedmay 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. Ifis the constant map fromtowith the Euclidean metric, thenis a single point hence is closed in