I came across a problem which looks like an exercise from Topology by Munkres, yet I can’t figure it out.
Let be a connected topological space. The product is also a topological space, and is connected as well. (Why? Because each “horizontal” fiber and each “vertical” fiber must lie within some connected component, but since they intersect, there is just one component.) However, if we remove the diagonal from the product, the connectedness may be lost. For example, is disconnected when is a line.
So far so easy. Now let’s take a quotient of , identifying each pair with the reordered pair . In other words, the symmetric group acts on by permutation of coordinates, and we take the space of orbits. This new space is denoted and is called the configuration space of .
It is in all examples that I can think of. Of course, this is not much evidence (for one thing, my thinking does not go beyond Hausdorff spaces). I’m happy to assume that is metrizable, in which case carries the Hausdorff metric . Here I’m using the notation for the metric to reduce the clutter.