Let be a topological space.
We say it is disconnected if we can find open and s.t. , and they are nonempty.
We say is connected if it is not disconnected.
The image of a continuous function on a connected set is connected.
The following are equivalent:
- is connected
- If then is an interval
- Every cts function where is discrete and is constant
A subspace of is disconnected iff there are open sets in s.t. , , and .
If is connected with equivalence relation then is connected.
(quotient map is cts)
Path-connected
A space is path-connected if for any , there is a cts function s.t. and .
Any path-connected space is connected. (Converse is not true).
Let be a function. If where and are closed and and are continuous, then is continuous.
Proof
Let be closed in .
which is a union of closed sets, hence closed.
Theorem
Let be open. Then is connected iff is path-connected.
Proof sketch
WLOG . Find . Let be path-connected component of . We show and are open in . Then they would disconnect it, unless .