Connectedness

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:

1. is connected
2. If then is an interval
3. 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 .