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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.