Math and science::Topology
Metric space. Function continuity
Function continuity between metric spaces
Let
For all
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
is continuous;- for all open
, [...]; - for all closed
, [...].
Informally: [2. can be rephrased as...].
Proof at the bottom of the back side.
This result motivates the definition of a topolgical space.