## locally path connected

33 examples (0.02 sec)
• Every locally simply connected space is also locally path-connected and locally connected.
• Being locally path connected, a manifold is path-connected if and only if it is connected.
• In particular, many authors require both spaces to be path-connected and locally path-connected.
• An open subset of a locally path-connected space is connected if and only if it is path-connected.
• In particular, they are locally simply connected, locally path connected, and locally connected.
• The space X has a universal cover if it is connected, locally path-connected and semi-locally simply connected.
• It is path-connected, locally path-connected and simply connected but not contractible.
• Since path connected spaces are connected, locally path connected spaces are locally connected.
• This means that is not locally path connected.
• It follows that an open connected subspace of a locally path connected space is necessarily path connected.
• The comb space is path connected but not locally path connected.
• This generalizes the earlier statement about R n and C n, each of which is locally path-connected.
• If H is a path-connected, locally path-connected, and semilocally simply connected group then it has a universal cover.
• Every finite space is locally path-connected since the set is a path-connected open neighborhood of x that is contained in every other neighborhood.
• Again suppose is a covering map and C (and therefore also X) is connected and locally path connected.
• Now suppose is a covering map and C (and therefore also X) is connected and locally path connected.
• A space is locally path connected if and only if for all open subsets U, the path components of U are open.
• Every hyperconnected space is both connected and locally connected (though not necessarily path-connected or locally path-connected).
• There are also local versions of these definitions: locally connected spaces and locally path connected spaces.
• More generally, any topological manifold is locally path-connected.