Locally Connected Space - Components and Path Components

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 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) 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 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 which 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,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 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, is connected and locally path connected, hence path connected, i.e., . That is, for a locally path connected space the components and path components coincide.

Examples

1. The set I × I (where I = ) 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.