In this topological space, V is a neighbourhood of p and it contains a connected open set (the dark green disk) that contains p.
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 (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.
Let be a topological space, and let be a point of
A space is called locally connected at  if every neighborhood of contains a connectedopen neighborhood of , that is, if the point 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 for example); and connectedness does not imply local connectedness (see the topologist's sine curve).
A space is called locally path connected at  if every neighborhood of contains a path connectedopen neighborhood of , that is, if the point 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.
A space is called connected im kleinen at  or weakly locally connected at  if every neighborhood of contains a connected neighborhood of , that is, if the point 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 is connected im kleinen at 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 is said to be path connected im kleinen at  if every neighborhood of contains a path connected neighborhood of , that is, if the point has a neighborhood base consisting of path connected sets.
A space that is locally path connected at is path connected im kleinen at 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.
For any positive integer n, the Euclidean space is locally path connected, thus locally connected; it is also connected.
Theorem — A space is locally connected if and only if it is weakly locally connected.
For the non-trivial direction, assume 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 be open in and let be a connected component of Let be an element of Then is a neighborhood of so that there is a connected neighborhood of contained in Since is connected and contains must be a subset of (the connected component containing ). Therefore is an interior point of Since was an arbitrary point of is open in Therefore, is locally connected.
Local connectedness is, by definition, a local property of topological spaces, i.e., a topological property P such that a space X possesses property P if and only if each point x in X admits a neighborhood base of sets that have property P. Accordingly, all the "metaproperties" held by a local property hold for local connectedness. In particular:
A space is locally connected if and only if it admits a base of (open) connected subsets.
The disjoint union of a family of spaces is locally connected if and only if each is locally connected. In particular, since a single point is certainly locally connected, it follows that any discrete space is locally connected. On the other hand, a discrete space is totally disconnected, so is connected only if it has at most one point.
Conversely, a totally disconnected space is locally connected if and only if it is discrete. This can be used to explain the aforementioned fact that the rational numbers are not locally connected.
A nonempty product space is locally connected if and only if each is locally connected and all but finitely many of the are connected.
The following result follows almost immediately from the definitions but will be quite useful:
Lemma: Let X be a space, and a family of subsets of X. Suppose that is nonempty. Then, if each is connected (respectively, path connected) then the union is connected (respectively, path connected).
Now consider two relations on a topological space X: for write:
if there is a connected subset of X containing both x and y; and
if there is a path connected subset of X containing both x and y.
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 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 of all points y such that is called the connected component of x. The Lemma implies that is the unique maximal connected subset of X containing x. Since
the closure of is also a connected subset containing x, it follows that 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., 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 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 of all points y such that is called the path component of x. As above, 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
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,y) 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 is connected and open, hence path connected, that is, That is, for a locally path connected space the components and path components coincide.
The set (where ) in the dictionaryorder topology has exactly one component (because it is connected) but has uncountably many path components. Indeed, any set of the form is a path component for each a belonging to I.
Let be a continuous map from to (which is in the lower limit topology). Since is connected, and the image of a connected space under a continuous map must be connected, the image of under must be connected. Therefore, the image of under must be a subset of a component of Since this image is nonempty, the only continuous maps from ' to are the constant maps. In fact, any continuous map from a connected space to a totally disconnected space must be constant.
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 containing x is called the quasicomponent of x.
can also be characterized as the intersection of all clopen subsets of X that contain x. Accordingly is closed; in general it need not be open.
Evidently for all  Overall we have the following containments among path components, components and quasicomponents at x:
If X is locally connected, then, as above, is a clopen set containing x, so and thus 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.
An example of a space whose quasicomponents are not equal to its components is a sequence with a double limit point. This space is totally disconnected, but both limit points lie in the same quasicomponent, because any clopen set containing one of them must contain a tail of the sequence, and thus the other point too.
The space is locally compact and Hausdorff but the sets and are two different components which lie in the same quasicomponent.
The Arens–Fort space is not locally connected, but nevertheless the components and the quasicomponents coincide: indeed for all points x.
^ abBjörn, Anders; Björn, Jana; Shanmugalingam, Nageswari (2016). "The Mazurkiewicz distance and sets that are finitely connected at the boundary". Journal of Geometric Analysis. 26: 873–897. arXiv:1311.5122. doi:10.1007/s12220-015-9575-9., section 2
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