Math and science::Topology

Metric space. Function continuity

Function continuity between metric spaces

Let $$X$$ and $$Y$$ be metric spaces and let $$f: X \to Y$$ be a function. We say that $$f$$ is continuous iff:

For all $$x_0 \in X$$ and for all $$\varepsilon > 0$$ there exists a [...] such that ([...] $$\implies$$ [...]).

Independence of metrics

The idea of continuity appears to depend on the notion of metric/distance as it is formulated using ε-balls. However, the following lemma reveals that this is not really so—all that is needed is the notion of open (or closed) subsets.

Equivance to function continuity

Let $$X$$ and $$Y$$ be metric spaces and let $$f : X \to Y$$ be a function. The the following three statements are equivalent:

1. $$f$$ is continuous;
2. for all open $$U \subseteq Y$$, [...];
3. for all closed $$V \subseteq Y$$, [...].

Informally: [2. can be rephrased as...].

Proof at the bottom of the back side.

This result motivates the definition of a topolgical space.