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 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 formula_1 (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 and first examples.
Let "X" be a topological space, and let "x" be a point of "X".
We say that "X" is locally connected at "x" if for every open set "V" containing "x" there exists a connected, open set "U" with formula_2. The space "X" is said to be locally connected if it is locally connected at "x" for all "x" in "X". Note that local connectedness and connectedness are not related to one another; a space may possess one or both of these properties, or neither.
By contrast, we say that "X" is weakly locally connected at "x (or connected im kleinen at "x) if for every open set "V" containing "x" there exists a connected subset "N" of "V" such that "x" lies in the interior of "N". An equivalent definition is: each open set "V" containing "x" contains an open neighborhood "U" of "x" such that any two points in "U" lie in some connected subset of "V". The space "X" is said to be weakly locally connected if it is weakly locally connected at "x" for all "x" in "X".
In other words, the only difference between the two definitions is that for local connectedness at "x" we require a neighborhood base of "open" connected sets containing "x", whereas for weak local connectedness at "x" we require only a neighborhood base of connected sets containing "x".
Evidently a space that is locally connected at "x" is weakly locally connected at "x". The converse does not hold (a counterexample, the broom space, is given below). On the other hand, it is equally clear that a locally connected space is weakly locally connected, and here it turns out that the converse does hold: a space that is weakly locally connected at all of its points is necessarily locally connected at all of its points. A proof is given below.
We say that "X" is locally path connected at "x" if for every open set "V" containing "x" there exists a path connected, open set "U" with formula_2. The space "X" is said to be locally path connected if it is locally path connected at "x" for all "x" in "X".
Since path connected spaces are connected, locally path connected spaces are locally connected. This time the converse does not hold (see example 6 below).
First examples.
Further examples are given later on in the article.
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 formula_14 a family of subsets of "X". Suppose that formula_15 is nonempty. Then, if each formula_16 is connected (respectively, path connected) then the union formula_17 is connected (respectively, path connected).
Now consider two relations on a topological space "X": for formula_18, 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 formula_21 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 formula_22 of all points "y" such that formula_23 is called the connected component of "x". The Lemma implies that formula_22 is the unique maximal connected subset of "X" containing "x". Since 
the closure of formula_22 is also a connected subset containing "x", it follows that formula_22 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., formula_27 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 formula_28 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 formula_29 of all points "y" such that formula_30 is called the "path component" of "x". As above, formula_29 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 formula_32 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,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 formula_33, 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", formula_22 is connected and open, hence path connected, i.e., formula_35. That is, for a locally path connected space the components and path components coincide.
Quasicomponents.
Let "X" be a topological space. We define a third relation on "X": formula_36 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 formula_37 containing "x" is called the quasicomponent of "x".
formula_37 can also be characterized as the intersection of all clopen subsets of "X" that contain "x". Accordingly formula_37 is closed; in general it need not be open.
Evidently formula_40 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, formula_22 is a clopen set containing "x", so formula_43 and thus formula_44. 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.
Sufficient conditions.
It is sufficient to show that the components of open sets are open. Let formula_50 be open in formula_51 and let formula_52 be a component of formula_53 Let formula_54 be an element of formula_55 Then formula_54 is an element of formula_50 so that there is a connected subspace formula_58 of formula_51 contained in formula_50 and containing a neighbourhood formula_61 of formula_54. Since formula_58 is connected and formula_58 contains formula_65 formula_58 must be a subset of formula_52 (the component containing formula_54). Therefore, the neighbourhood formula_61 of formula_54 is a subset of formula_71 which shows that formula_54 is an interior point of formula_55 Since formula_54 was an arbitrary point of formula_71 formula_52 is open in formula_77 Therefore, formula_51 is locally connected.
A certain infinite union of decreasing broom spaces is an example of a space that is weakly locally connected at a particular point, but not locally connected at that point.
A first-countable Hausdorff space formula_79 is locally path-connected if and only if formula_80 is equal to the final topology on formula_51 induced by the set formula_82 of all continuous paths formula_83