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:
“Resorts advertised for waitresses, specifying that they “must appear in short clothes or no engagement.” Below a Gospel Guide column headed, “Where our Local Divines Will Hang Out Tomorrow,” was an account of spirited gun play at the Bon Ton. In Jeff Winney’s California Concert Hall, patrons “bucked the tiger” under the watchful eye of Kitty Crawhurst, popular “lady” gambler.”
—Administration in the State of Colo, U.S. public relief program (1935-1943)