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.

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 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. 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 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 (not necessarily open) 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. However, if a space is connected im kleinen at each of its points, it is locally connected. A space X is said to be path connected im kleinen at x if every neighborhood of x contains a path connected (not necessarily open) 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

A first-countable Hausdorff space (X, \tau) is locally path-connected if and only if \tau is equal to the final topology on X induced by the set of all continuous paths

Properties

For the non-trivial direction, assume X is weakly locally connected. To show it is locally connected, it is enough to show that the connected components of open sets are open. Let U be open in X and let C be a connected component of U. Let x be an element of C. Then U is a neighborhood of x so that there is a connected neighborhood V of x contained in U. Since V is connected and contains x, V must be a subset of C (the connected component containing x). Therefore x is an interior point of C. Since x was an arbitrary point of C, C is open in X. Therefore, X is locally connected.

Components and path components

The following result follows almost immediately from the definitions but will be quite useful: Lemma: Let X be a space, and {Y_i} a family of subsets of X. Suppose that is nonempty. Then, if each Y_i is connected (respectively, path connected) then the union is connected (respectively, path connected). Now consider two relations on a topological space X: for x,y \in X, write: Evidently both relations are reflexive and symmetric. Moreover, if x and y are contained in a connected (respectively, path connected) subset A and y and z are connected in a connected (respectively, path connected) subset B, then the Lemma implies that A \cup B is a connected (respectively, path connected) subset containing x, y and z. Thus each relation is an equivalence relation, and defines a partition of X into equivalence classes. We consider these two partitions in turn. For x in X, the set C_x of all points y such that is called the connected component of x. The Lemma implies that C_x is the unique maximal connected subset of X containing x. Since the closure of C_x is also a connected subset containing x, it follows that C_x is closed. If X has only finitely many connected components, then each component is the complement of a finite union of closed sets and therefore open. In general, the connected components need not be open, since, e.g., there exist totally disconnected spaces (i.e., C_x = {x} for all points x) that are not discrete, like Cantor space. However, the connected components of a locally connected space are also open, and thus are clopen sets. It follows that a locally connected space X is a topological disjoint union \coprod C_x of its distinct connected components. Conversely, if for every open subset U of X, the connected components of U are open, then X admits a base of connected sets and is therefore locally connected. Similarly x in X, the set PC_x of all points y such that is called the path component of x. As above, PC_x is also the union of all path connected subsets of X that contain x, so by the Lemma is itself path connected. Because path connected sets are connected, we have for all x \in X. 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 which is closed but not open. A space is locally path connected if and only if for all open subsets U, the path components of U are open. Therefore the path components of a locally path connected space give a partition of X into pairwise disjoint open sets. It follows that an open connected subspace of a locally path connected space is necessarily path connected. Moreover, if a space is locally path connected, then it is also locally connected, so for all x \in X, C_x is connected and open, hence path connected, that is, C_x = PC_x. That is, for a locally path connected space the components and path components coincide.

Examples

Quasicomponents

Let X be a topological space. We define a third relation on X: if there is no separation of X into open sets A and B such that x is an element of A and y is an element of B. This is an equivalence relation on X and the equivalence class QC_x containing x is called the quasicomponent of x. QC_x can also be characterized as the intersection of all clopen subsets of X that contain x. Accordingly QC_x is closed; in general it need not be open. Evidently for all x \in X. Overall we have the following containments among path components, components and quasicomponents at x: If X is locally connected, then, as above, C_x is a clopen set containing x, so and thus QC_x = C_x. Since local path connectedness implies local connectedness, it follows that at all points x of a locally path connected space we have Another class of spaces for which the quasicomponents agree with the components is the class of compact Hausdorff spaces.

Examples