Locally Connected Space - Definitions and First Examples

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 . The space X is said to be locally connected if it is locally connected at x for all x in X.

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, whereas for weak local connectedness at x we require only a base of neighborhoods of 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 . 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 is connected and locally connected.

2. The subspace of the real line is locally 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 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. Let X be a countably infinite set endowed with the cofinite topology. Then X is locally connected (indeed, hyperconnected) but not locally path connected.

Further examples are given later on in the article.