Locally connected space

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 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 x ∈ U ⊂ V . 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 which 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 which 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 x ∈ U ⊂ V . 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

1. For any positive integer n, the Euclidean space R n is locally path connected, thus locally connected; it is also connected.
2. The subspace [ 0 , 1 ] ∪ [ 2 , 3 ] of the real line R 1 is locally path connected but not connected.
3. The topologist's sine curve is a subspace of the Euclidean plane which is connected, but not locally connected.
4. The space Q of rational numbers endowed with the standard Euclidean topology, is neither connected nor locally connected.
5. The comb space is path connected but not locally path connected.
6. A countably infinite set endowed with the cofinite topology is locally connected (indeed, hyperconnected) but not locally path connected.

Further examples are given later on in the article.

Properties

1. 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 which have property P. Accordingly, all the "metaproperties" held by a local property hold for local connectedness. In particular:
2. A space is locally connected if and only if it admits a base of connected subsets.
3. The disjoint union ∐ i X i of a family { X i } of spaces is locally connected if and only if each X i 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.
4. 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.

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 ⋂ i Y i is nonempty. Then, if each Y i is connected (respectively, path connected) then the union ⋃ i Y i is connected (respectively, path connected).

Now consider two relations on a topological space X: for x , y ∈ X , write:

x ≡ c y if there is a connected subset of X containing both x and y; and x ≡ p c y 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 A ∪ 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 y ≡ c x 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) which 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 ∐ 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 P C x of all points y such that y ≡ p c x is called the path component of x. As above, P C x is also the union of all path connected subsets of X which contain x, so by the Lemma is itself path connected. Because path connected sets are connected, we have P C x ⊂ C x 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 C ∖ U , 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 locally path connected, hence path connected, i.e., C x = P C x . That is, for a locally path connected space the components and path components coincide.

Examples

1. The set I × I (where I = [0,1]) in the dictionary order topology has exactly one component (because it is connected) but has uncountably many path components. Indeed, any set of the form {a} × I is a path component for each a belonging to I.
2. Let f be a continuous map from R to R_ℓ (R in the lower limit topology). Since R is connected, and the image of a connected space under a continuous map must be connected, the image of R under f must be connected. Therefore, the image of R under f must be a subset of a component of R_ℓ. Since this image is nonempty, the only continuous maps from R to R_ℓ, are the constant maps. In fact, any continuous map from a connected space to a totally disconnected space must be constant.

Quasicomponents

Let X be a topological space. We define a third relation on X: x ≡ q c y 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 Q C x containing x is called the quasicomponent of x.

Q C x can also be characterized as the intersection of all clopen subsets of X which contain x. Accordingly Q C x is closed; in general it need not be open.

Evidently C x ⊆ Q C x for all x in X. Overall we have the following containments among path components, components and quasicomponents at x:

P C x ⊆ C x ⊆ Q C x .

If X is locally connected, then, as above, C x is a clopen set containing x, so Q C x ⊆ C x and thus Q C 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

P C x = C x = Q C x .

Examples

1. An example of a space whose quasicomponents are not equal to its components is a countable set, X, with the discrete topology along with two points a and b such that any neighbourhood of a either contains b or all but finitely many points of X, and any neighbourhood of b either contains a or all but finitely many points of X. The point a lies in the same quasicomponent of b but not in the same component as b.
2. The Arens–Fort space is not locally connected, but nevertheless the components and the quasicomponents coincide: indeed Q C x = C x = { x } for all points x.

More on local connectedness versus weak local connectedness

Theorem

Let X be a weakly locally connected space. Then X is locally connected.

Proof

It is sufficient to show that the components of open sets are open. Let U be open in X and let C be a component of U. Let x be an element of C. Then x is an element of U so that there is a connected subspace A of X contained in U and containing a neighbourhood V of x. Since A is connected and A contains x, A must be a subset of C (the component containing x). Therefore, the neighbourhood V of x is a subset of C. Since x was arbitrary, we have shown that each x in C has a neighbourhood V contained in C. This shows that C is open relative to U. Therefore, X is locally connected.

A certain infinite union of decreasing broom spaces is an example of a space which is weakly locally connected at a particular point, but not locally connected at that point.