Long ago we talked about topological spaces being connected, which meant we couldn’t partition the space into two disjoint non-empty open sets. There is a stronger notion that we’ll need when working with the fundamental group called *path connectedness*. We say that a space is path connected if for every pair of points there exists a path in the space, , such that and ; we can connect every pair of points with a path.

If a space is not path connected, it can be decomposed into a collection of disjoint subsets with *are* path connected. These subsets are called the *path components* of the space. More specifically, path connectedness is an equivalence relation on the space and the path components are the equivalence classes. Clearly every point is path connected to itself (reflexivity); if and are connected by path , then the reverse path connects and (symmetry); and the concatenation operation from last time gives us transitivity.

We said that path connectedness was a stronger condition than connectedness. This means that path connectedness implies regular ol’ connectedness. To see this, suppose that is path connected. Let be connected by path . Let be an open set containing , and an open set containing . Now consider the preimages and . Since is a path it is continuous, so these are two open subsets of [0, 1]. We must have since and . We know that [0, 1] is connected, however. This implies that must be connected (otherwise their preimages would be disjoint open sets covering [0, 1]). Since and were arbitrary, we can generalize to get that is connected.

Notice that this also means that a continuous path must stay inside of one path component.

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