Note that by the unique lifting property, if f is not the identity and C is path connected, then f has no fixed points.

The comb space is path connected but not locally path connected.

Then X is locally connected (indeed, hyperconnected) but not locally path connected.

There are also paths connected to the route throughout the area.

The set of invertible elements in L(H) is path connected.

Like all finite topological spaces, S is locally path connected.

Because of this, different terminology is used; spaces with this property are said to be path connected.

The space is locally connected and path connected, while not arc connected.

If a space is connected or path connected, then so are all its quotient spaces.

As a topological space it is compact, contractible, path connected and locally path connected.