Local Connectedness
A topological space is said to be locally connected at a point x if every neighbourhood of x contains a connected open neighbourhood. It is locally connected if it has a base of connected sets. It can be shown that a space X is locally connected if and only if every component of every open set of X is open. The topologist's sine curve is an example of a connected space that is not locally connected.
Similarly, a topological space is said to be locally path-connected if it has a base of path-connected sets. An open subset of a locally path-connected space is connected if and only if it is path-connected. This generalizes the earlier statement about Rn and Cn, each of which is locally path-connected. More generally, any topological manifold is locally path-connected.
Read more about this topic: Connected Space
Famous quotes containing the word local:
“America is the world’s living myth. There’s no sense of wrong when you kill an American or blame America for some local disaster. This is our function, to be character types, to embody recurring themes that people can use to comfort themselves, justify themselves and so on. We’re here to accommodate. Whatever people need, we provide. A myth is a useful thing.”
—Don Delillo (b. 1926)