The definition of a Hausdorff space says that points can be separated by neighborhoods. It turns out that this implies something which is seemingly stronger: in a Hausdorff space every pair of disjoint compact sets can also be separated by neighborhoods, in other words there is a neighborhood of one set and a neighborhood of the other, such that the two neighborhoods are disjoint. This is an example of the general rule that compact sets often behave like points.
The definition of a Hausdorff space says that points can be separated by neighborhoods It turns out that this implies something which is seemingly stronger in a Hausdorff space every pair of disjoint compact sets can also be separated by neighborhoods in other words there is a neighborhood of one set and a neighborhood of the other such that the two neighborhoods are disjoint This is an example of the general rule that compact sets often behave like points