I’ve been reviewing some basic general topology as of late. I will post some of this material here. Apologies to readers who prefer more advanced topics; my current focus is on foundational material.

Often, we’d like to prove that a given subset of a topological space has some given property, e.g. that it is open or closed. In many cases, however, the big space may not be easily understandable, but local pieces of it may be. For instance, might be a manifold, and we might not know what the global structure of is, but we do know that is *locally* homeomorphic (or diffeomorphic) to a ball in . So we need a way to go from local results to global results.

**Proposition 1** * Let be an open cover of the topological space . Suppose and is open in for each . Then is open in . *

So openness is a local property.

This is the easy result. Indeed, since is an open set, each is open in . But

so that is a union of open sets, hence open.

Similarly, we can deduce the corresponding result for closed sets:

**Corollary 2** * Suppose is an open cover of . Let . Suppose is closed in for each . Then is closed in . *

Indeed, this follows from the previous result, with replaced with .

However, the analogous result is no longer true if we look at closed covers. Consider for instance the closed cover of by vertical lines. The set , defined as the intersection of the graph with the upper right quadrant, is not closed, though its intersection with each vertical line is closed (in fact, is a point). So we need something more. The problem, as we will see, is that there are too many lines. (more…)