Applying connectedness

This article is about how one can use the point set-topological fact that a given topological space (possibly, with a lot of additional structure) is connected.

It is important to distinguish being connected from being path-connected, which is in general a stronger condition. For locally path-connected spaces (and in particular for manifolds) the two conditions are equivalent. However, it often happens that connectedness, though a weaker formulation than path-connectedness, is more useful for certain purposes. Also refer the article on applying path-connectedness.

Group actions and transitivity
One of the frequent applications of connectedness is to somehow show that all points of the space look similar. The typical setup is that we have a subgroup of the self-homeomorphism group of the topological space, or equivalently, a group action on the topological space, and we want to show that this action is transitive. In other words, if $$X$$ is the topological space and $$G$$ is a group acting on $$X$$, we want to prove that for any $$x,y \in X$$ there exists $$g \in G$$ sending $$x$$ to $$y$$.

The hypothesis of connectedness is typically used in the following way: we combine with the local structure of the space:


 * We show that for every point, there exists a small neighbourhood of the point such that the group acts transitivity in that neighbourhood
 * Thus, we show that every orbit under the group action is an open set
 * This gives a partition of the whole space into open sets. Since the space is connected, there exists exactly one orbit, proving that the action is transitive.

Another way of viewing this is that connectedness enables us to go from a local behaviour to a global behaviour.

Some examples of this:


 * Connected manifold implies homogeneous
 * Any connected smooth manifold is homogeneous under the action of the self-diffeomorphism group