Locally connected space

In topology and other branches of mathematics, a topological space X is locally connected if every point admits a neighbourhood basis consisting of open connected sets.

As a stronger notion, the space X is locally path connected if every point admits a neighbourhood basis consisting of open path connected sets.

Indeed, the study of these properties even among subsets of Euclidean space, and the recognition of their independence from the particular form of the Euclidean metric, played a large role in clarifying the notion of a topological property and thus a topological space.

This led to a rich vein of research in the first half of the twentieth century, in which topologists studied the implications between increasingly subtle and complex variations on the notion of a locally connected space.

As an example, the notion of local connectedness im kleinen at a point and its relation to local connectedness will be considered later on in the article.

In the latter part of the twentieth century, research trends shifted to more intense study of spaces like manifolds, which are locally well understood (being locally homeomorphic to Euclidean space) but have complicated global behavior.

By this it is meant that although the basic point-set topology of manifolds is relatively simple (as manifolds are essentially metrizable according to most definitions of the concept), their algebraic topology is far more complex.

From this modern perspective, the stronger property of local path connectedness turns out to be more important: for instance, in order for a space to admit a universal cover it must be connected and locally path connected.

It follows, for instance, that a continuous function from a locally connected space to a totally disconnected space must be locally constant.

In fact the openness of components is so natural that one must be sure to keep in mind that it is not true in general: for instance Cantor space is totally disconnected but not discrete.

has a neighborhood base consisting of connected open sets.

contains a path connected open neighborhood of

has a neighborhood base consisting of path connected open sets.

The converse does not hold (see the lexicographic order topology on the unit square).

contains a connected (not necessarily open) neighborhood of

has a neighborhood base consisting of connected sets.

The converse does not hold, as shown for example by a certain infinite union of decreasing broom spaces, that is connected im kleinen at a particular point, but not locally connected at that point.

contains a path connected (not necessarily open) neighborhood of

has a neighborhood base consisting of path connected sets.

The converse does not hold, as shown by the same infinite union of decreasing broom spaces as above.

[11][better source needed] A first-countable Hausdorff space

The following result follows almost immediately from the definitions but will be quite useful: Lemma: Let X be a space, and

write: Evidently both relations are reflexive and symmetric.

is the unique maximal connected subset of X containing x.

In general, the connected components need not be open, since, e.g., there exist totally disconnected spaces (i.e.,

[22] It follows that a locally connected space X is a topological disjoint union

However the closure of a path connected set need not be path connected: for instance, the topologist's sine curve is the closure of the open subset U consisting of all points (x,sin(x)) with x > 0, and U, being homeomorphic to an interval on the real line, is certainly path connected.

Moreover, the path components of the topologist's sine curve C are U, which is open but not closed, and

[24] Therefore the path components of a locally path connected space give a partition of X into pairwise disjoint open sets.

can also be characterized as the intersection of all clopen subsets of X that contain x.