Connectedness and Paths

Connectedness

A topological space is called connected if there do not exist two disjoint nonempty open sets and such that .

e.g. The set of rational numbers in the standard Euclidean topology is not connected because with

Proposition

Let be endowed with the standard Euclidean topology. A set is connected if and only if is an interval.

Theorem

Let and be topological spaces. If is continuous and is connected, then is connected, i.e. the continuous image of a connected set is connected.

Corollary

Let be a connected topological space and a continuous function, where is endowed with the standard topology. If takes the values and , then takes all the values between and .

Proof By the given conditions, is connected and hence it is an interval. Since , any value between and must be in .

Proposition

The only subset of a connected topological space which is both open and closed is the empty set and itself.

Proof

Path Connectedness

Path

Let be a topological space. A path in joining two points is a continuous function such that and . If the start point and end point coincide, we call it a loop.

Path Connectedness

A topological space is called path connected if for every pair of points can be joined by a path in .

Theorem

A path connected topological space is connected. In general, connected space is not necessarily path-connected.

Proof If is not connected, then there exist disjoint nonempty open sets , in with . Let and . Since is path-connected, there is a path joining to . By the continuity of , and are disjoint nonempty open sets in with Therefore is not connected, which is a contradiction.

Proposition

If is an open set in , then is connected if and only if is path connected.

Proof It suffices to show that if is nonempty connected, then it is path connected. Fix some , we define . Clearly because . By the proposition, it is enough to show that is both open and closed in , which implies that , hence is path connected. For any , since is open, there is a open ball contained in . Let be the path from to . Note that for all , we can define a path from to by connecting and , so , which shows that is open in . Now we show that is closed in . Let be a sequence converging to . As is open, there is some open ball , and for all sufficiently large . Similarly by concatenation of paths, we can define a path joining to and then to , hence . This shows that is closed in . Therefore, is path connected.

Locally Path Connected

A topological space is locally path-connected if for any point and any open neighborhood of , there exists a path-connected open neighborhood containing .

e.g. Consider . Clearly it is connected but not locally path-connected because for any point , any open neighborhood of contains points of the form for , which cannot be connected by a path in .

Proposition

If a topological space is both connected and locally path-connected, then it must be path-connected.