Real Analysis – Why Not Use Closed Covers to Define Compactness of Metric Space?

compactnessgeneral-topologymetric-spacesreal-analysis

I'm a beginner in metric space. So many books I've read, there is only the notion of open covers. I want to know why do we worry about open covers to define the compactness of metric spaces and why don't we use closed covers? What is the problem in defining closed cover of a set? Can we use the alternative definition of compactness: "Every closed cover has a finite subcover"?

Best Answer

It is important to understand that, although definitions often look arbitrary, they never are. Mathematical objects are intended to model something, and you can't understand why the definition is the way it is until you understand what it is trying to model. The question you asked is exactly the right one: why is it defined this way and not some other way? What is it trying to model?

(For example, why does a topology say that arbitrary unions of open sets are open, but infinite intersections of open sets might not be? It's because topology is intended to be an abstraction of certain properties of the line and the plane, and open sets are intended to be a more general version of open intervals of the line and open discs in plane, and that is how the intervals and discs behave.)

This case is similar. Mathematicians noticed that there are certain sorts of “well-behaved” subsets of the line and of metric spaces in general. For example:

  1. A continuous function is always uniformly continuous — if and only if its domain is well-behaved in this way
  2. A continuous real-valued function is always bounded — if and only if its domain is well-behaved in this way
  3. If $f$ is a continuous real-valued function on some domain, there may be some $m$ at which $f$ is maximized: $f(x) ≤ f(m)$ for all $x$. This is true of all such $f$ if and only if the domain is well-behaved in this way
  4. Every sequence of points from a subset of $\Bbb R^n$ contains a convergent subsequence — if and only if the subset is well-behaved in this way

and so on. It took mathematicians quite a long time to understand this properly, but the answer turned out to be that the "well-behaved" property is compactness. There are several equivalent formulations of it, including the open cover formulation you mentioned.

In contrast, the alternative property you propose, with closed covers, turns out not to model anything interesting, and actually to be trivial, as the comments point out. It ends nowhere. But even if it ended somewhere nontrivial, it would be a curiosity, of not much interest, unless it had started from a desire to better understand of something we already wanted to understand. It's quite easy to make up new mathematical properties at random, and to prove theorems about those properties, and sometimes it might seem like that is what we are doing. But we never are.

Properly formulated, compactness turns out to be surprisingly deep. Before compactness, mathematics already had an idea of what a finite set was. Finite sets are always discrete, but not all discrete sets are finite.
Compactness is the missing ingredient: a finite set is one that is both discrete and compact. With the discovery of compactness, we were able to understand finiteness as a conjunction of two properties that are more fundamental! Some of the properties we associate with finiteness actually come from discreteness; others come from compactness. (Some come from both.) Isn't that interesting?

And formulating compactness correctly helps us better understand the original space, $\Bbb R^n$ and metric spaces in general. Once we get compactness right, we see that the properties of "well-behaved" sets I mentioned above are not true of all compact spaces; metric spaces are special in several ways, which we didn't formerly appreciate.

Keep asking these questions. Every definition is made for a reason.

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