But why is Compactness important?

Thu, 15 Aug 2019 01:34:25 +0530

In my last post I touched upon the intuition behind topological compactness. We as engineers often hear about the word ‘compact’ as a soft gatekeeping tool from doing serious mathematics. In this post we see why the understanding is very important for doing any mathematics, especially as engineers.

Recall the open cover definition of a compact set as the existence of a finite open subcover for any given open cover of the set. Note that the key words here are that the subcover is going to be finite, no matter which infinite cover we begin with. Let us keep that in mind. Now we often hear about compactness being a more subtle form of finiteness, or compactness abstracting the idea of being finite, or the two being related. The open cover definition lets us understand that slightly better. Open sets, being the building blocks of topology, give us a way to interact with a specific class of sets in a finite manner ¹. It is this class of sets that we call compact.

Compactness

Sat, 10 Aug 2019 15:44:20 +0530

This is a non-mathematical note on what I understand about compactness and what it means for a set or a space to be compact. The open cover definition is one that can be found in any textbook, but what does it mean for a set to be compact? Why are such sets called compact? And how do compact sets differ from those which are not?

I think the terminology here is very carefully chosen. For once, mathematicians came up with a term that paints a succinct, abstract picture. To understand compactness,let us look at two other properties: limit points, and closed sets.

Real Analysis on Omanshu Thapliyal

But why is Compactness important?

Compactness