O Sweet Mr Math

wherein is detailed Matt's experiences as he tries to figure out what to do with his life. Right now, that means lots of thinking about math.

Wednesday, March 28, 2012

8:10 PM

Last time we defined compact sets. Unfortunately, while open sets and closed sets have an intuitive meaning to me, and that intuitive meaning is backed up by the formal definition, with compact sets I only have the formal definition to work from. So, a set E in a metric space X is a compact set if for any collection of open sets, {Gα}, such that E is a subset of the union of G, there is a finite subcollection of the open sets such that E is a subset of the subcollection.

That feels really messy and nebulous to me. The important thing is that every open cover {Gα} of E is required to have a finite number of sets which is also an open cover of E. E is just a single set, and it's easy to think of different possibilities for E, and then ask which possibilities are compact. But "any open cover" is a really broad concept. I think the place to start is with lots of small neighborhoods of individual points. Working with my "metric space visualized as a sheet of paper" model from last time, it's straight forward to visualize E as some group of shapes, potentially including enclosed spaces, curved lines including all the points on the line, and individual points. Then any open cover is going to look like lots of little tiny overlapping circles which cover all of E. Start with one circle for each point in E, so if E contains an infinite number of points then there will be an infinite number of sets in the open cover. Then start removing redundant sets, and see if you can get down to a finite number of sets.

With that unwieldy definition, the goal now is to explore some properties of compact sets in the hope that we can get a better handle on them. Our first property is that a compact set is compact regardless of the space that contains it.

We looked a bit at open sets, and how openness is dependent on the underlying space. Our example was an open segment on the real number line. If the space is the real number line, with the standard definition for distance, then the segment is an open set. But if the space is the complex numbers, with the same definition for distance, then the segment is not an open set. Compact sets are better behaved.

Suppose E is a compact set in some space Y, and that Y is a subset of some larger space X. Since E is a subset of X, we can pick any open cover of E in X. In fact, we can consider every open cover of E in X. So {Gα} is a collection of open sets in X, and E is a subset of the union of all the sets in the collection. The collection can be infinite for now, but we will reduce it to a finite collection. For each set in the collection, take the intersection of that set with the set Y. We know that Gα∩Y is an open set in Y. Every point in E which is in Gα is also in Gα∩Y. Therefore the collection {Gα∩Y} is an open cover of E in Y. Since E is compact in Y, a finite subcollection of that open cover is also an open cover of E. If Gα∩Y (for some particular value of α) is in the finite open cover, then keep Gα as part of an open cover of E in X. Since the collection of sets in Y is finite, the collection of sets in X is the same size, and also finite, and therefore E is compact in X. If E⊂Y⊂X and E is compact in Y, then E is compact in X.

Suppose, instead, that E is a compact set in some space X. Furthermore, suppose E is a subset of Y, which is some subset of X. (In other words, we're considering the reverse of the previous situation. E⊂Y⊂X and E is compact in X.) Take some open cover of E in Y. (Again, this works with any possible open cover.) If V is one of the sets in the open cover, V is open relative to Y, but is not necessarily open relative to X. However, there exists a set G which is open relative to X, such that V=G∩Y. So, for every set in the open cover in Y, we can find an open set in X. This collection of open sets in X is an open cover of E in X. Since E is compact in X, a finite subcollection of these sets is also an open cover of E. Take the corresponding open sets in Y, and we now have a finite open cover of E in Y. Therefore, E is a compact set in Y.

This means, essentially, that once an compact set, always a compact set. If E is a compact set in any metric space which contains E, it is a compact set in every space which contains E. Since subsets of metric spaces are themselves metric spaces, and because E is compact regardless of what space may contain it, it makes sense to refer to E as a compact space. This makes it unlike open sets or closed sets, because they are only open or closed relative to a particular space. Every set is both open and closed relative to itself, so it doesn't really make sense to refer to an open or closed space.

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