Best of all is being compact and Hausdorff . Being Hausdorff essentially means that there are enough open sets to stop the points in X getting 'tangled up' with one another. Often, it turns out that some of the sets in an open cover are redundant. If we can remove sets from a cover and what's left behind still covers X, we call the smaller collection a subcover of the larger one. Given this definition, it turns out that R and Q are not compact, but that every closed and bounded subset of R is compact.
Compact spaces are very popular with topologists, because they're very easy to reason about.
If X is one of the standard spaces used in complicated mathematics it probably isn't compact, and that means it may be quite hard to reason about its properties. What we try to do, therefore, is find some space K which is very similar to X, but which is easier to reason about. For example, every set is dense in its closure. This limits which spaces X can be treated in this way, but it's only a minor problem because just about all of the most useful spaces can be handled using this approach. Because the endpoints are missing, we can't complete the task.