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
, the preimage is open; - for all closed
, the preimage is closed.
Informally: 'the preimage of an open set is open'.
Proof at the bottom of the back side.
This result motivates the definition of a topolgical space.
The transition to Topology
With function continuity as the target concept, what is the minimum set of ideas requiring construction before function continuity can be formulated?
Without introducing the idea of a metric function, a space which we will call a topological space will be defined as being a
set
- Arbitrary unions of open subsets are open.
- Finite intersections of open subsets are open.
and are open.
This formulation could equally have been done with closed rather than open subsets.
After this restructuring, a function will be defined to be continuous iff 'the preimage of an open set is open'.
Equivalence of Continuity, proof
Below is a proof for metric spaces of the equivalence of the three statements above regarding continuity. Below again is an attempt to visualize the line of reasoning.
The main 'trick' of the proof is to iterate through each
Context
![](_topology_mindmap.png)