Locally path-connected

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

Background

Throughout the history of topology, connectedness and compactness have been two of the most
widely studied topological properties. 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. However, whereas the structure of ''compact'' subsets of Euclidean space was understood quite early on via the Heine–Borel theorem, ''connected'' subsets of $\R^n$ (for ''n'' > 1) proved to be much more complicated. Indeed, while any compact Hausdorff space is locally compact, a connected space—and even a connected subset of the Euclidean plane—need not be locally connected (see below).

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 weak local connectedness 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. Local path connectedness will be discussed as well.

A space is locally connected if and only if for every open set ''U'', the connected components of ''U'' (in the subspace topology) are open. 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.

Definitions

Let $X$ be a topological space, and let $x$ be a point of $X.$

A space $X$ is called locally connected at $x$Munkres, p. 161 if every neighborhood of $x$ contains a connected ''open'' neighborhood of $x$, that is, if the point $x$ has a neighborhood base consisting of connected open sets. A locally connected space is a space that is locally connected at each of its points.

Local connectedness does not imply connectedness (consider two disjoint open intervals in $\R$ for example); and connectedness does not imply local connectedness (see the topologist's sine curve).

A space $X$ is called locally path connected at $x$ if every neighborhood of $x$ contains a path connected ''open'' neighborhood of $x$, that is, if the point $x$ has a neighborhood base consisting of path connected open sets. A locally path connected space is a space that is locally path connected at each of its points.

Locally path connected spaces are locally connected. The converse does not hold (see the lexicographic order topology on the unit square).

Connectedness im kleinen

A space $X$ is called connected im kleinen at $x$ or weakly locally connected at $x$ if every neighborhood of $x$ contains a connected neighborhood of $x$, that is, if the point $x$ has a neighborhood base consisting of connected sets. A space is called weakly locally connected if it is weakly locally connected at each of its points; as indicated below, this concept is in fact the same as being locally connected.

A space that is locally connected at $x$ is connected im kleinen at $x.$ 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.Steen & Seebach, example 119.4, p. 139Munkres, exercise 7, p. 162 However, if a space is connected im kleinen at each of its points, it is locally connected.Willard, Theorem 27.16, p. 201

A space $X$ is said to be path connected im kleinen at $x$, section 2 if every neighborhood of $x$ contains a path connected neighborhood of $x$, that is, if the point $x$ has a neighborhood base consisting of path connected sets.

A space that is locally path connected at $x$ is path connected im kleinen at $x.$ The converse does not hold, as shown by the same infinite union of decreasing broom spaces as above. However, if a space is path connected im kleinen at each of its points, it is locally path connected.

First examples

# For any positive integer ''n'', the Euclidean space $\R^n$ is locally path connected, thus locally connected; it is also connected.
# More generally, every locally convex topological vector space is locally connected, since each point has a local base of convex (and hence connected) neighborhoods.
# The subspace S = ,1\cup ,3/math> of the real line $\R^1$ is locally path connected but not connected.
# The topologist's sine curve is a subspace of the Euclidean plane that is connected, but not locally connected.Steen & Seebach, pp. 137–138
# The space $\Q$ of r