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How to understand the concept of compact space

Hello,

I am learning some analysis on my own and

what is the motivation to consider compactness?

eg. I do not understand why having the property of reducing a cover to a finite cover is important.

I'm sure there is a good answer to this and I'm interested to hear your replies!

--Alex

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    $\begingroup$ You might consult the earlier question: mathoverflow.net/questions/25977 . $\endgroup$
    – Robin Chapman
    Commented Jun 30, 2010 at 6:12
  • $\begingroup$ Alex, doesn't your book report a number of theorems about compactness? Try finding counterexamples when the hypothesis of compactnes is dropped. $\endgroup$
    – Pietro Majer
    Commented Jun 30, 2010 at 6:42
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    $\begingroup$ Your question is answered by Wikipedia, so look there: en.wikipedia.org/wiki/Compactness, which also explains how the concept developed. A quote from Frechet (who first defined metric spaces): "We have already pointed out and will recognize throughout this book the importance of compact sets. All those concerned with general analysis have seen that it is impossible to do without them." (For Frechet, compactness meant what we'd call sequential compactness, which for your analysis reading is equivalent to the open cover version.) $\endgroup$
    – KConrad
    Commented Jun 30, 2010 at 7:12
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    $\begingroup$ As this query heads for foreclosure, I'll add something not covered in the thread Robin cited. For visualing compactness there is the Heine-Borel theorem saying what it means for sets in ordinary geometric space, and in some sense that is that. However, this is misleading, in the sense that the real force of compactness comes not from intuition but from decades of accumulated experience; it is a tameness hypothesis (similar to measurable, separable, or Noetherian) that appears extremely often as a natural dividing line or organizing principle. $\endgroup$
    – T..
    Commented Jun 30, 2010 at 7:18
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    $\begingroup$ I just noticed that the OP self-identifies as an undergraduate at Washington University in Saint Louis (whereas the phrasing of the question suggested to me that he was without university affiliation). In that case: you should ask these kinds of questions of the professors in the math department! That's what you're paying for (and, in part, what they're getting paid for). $\endgroup$
    – Pete L. Clark
    Commented Jun 30, 2010 at 7:33

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First of all, the term "compact" is perhaps well-suited because any open cover has a finite ("small") sub-cover. But more generally, compactness generalizes certain properties of closed and bounded intervals $[a,b]$ in $\mathbb{R}$ to abstract topological spaces (and closed and bounded subsets of $\mathbb{R}^n$ are compact by the Bolzano–Weierstrass theorem; the Heine–Borel theorem applies to metric spaces), such as sequential compactness -- staying "trapped" within the set.

In particular, various notions of compactness can be helpful for dealing with e.g. function spaces. They again generalizes properties of the real line -- for instance, functions from $X$ (a compact metric space) to $Y$ (an arbitrary metric space) are uniformly continuous. The continuous image of a compact set is compact, just as in $\mathbb{R}$. Finally, we have the Stone-Weierstrass theorem for compact Hausdorff spaces and algebras of continuous functions, generalizing the Weierstrass approximation theorem for polynomials defined on closed intervals. There are many other theorems for abstract topological spaces where compactness is important.

Of course, these theorems apply to $\mathbb{R}^n$ as well!

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You might want to look at the answers for this question: Applications of compactness

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The following is paraphrased from Lee's introduction to topological manifolds:

"A Fundamental fact about continuous functions is the extreme value theorem: A continuous real valued function on a closed bounded subset of $\mathbb{R}$ attains its maximum and minimum values.

The proof of this result hinges on the compactness of closed bounded subsets of $\mathbb{R}$.

This indicates that one might be able to formulate the extreme value theorem in more general situations, and it might be fruitful to study the notion of compactness further."

My guess is that there is no really simple way to motivate the modern definition compactness. Does anyone know when the modern definition appeared? Im guessing that the definition of sequential compactness appeared before the definition of compactness

Edit: The History behind the definition of compactness is given on the wikipedia page.

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