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I can easily understand the notion of compactness for Euclidean spaces $\mathbb{R}^n$. But, I am having a hard time for more general spaces. Precisely, my question is:

Whoever first defined compactness using the open cover definition, how did he arrive at this definition?

I have read a lot of similar posts, like this and this, but beyond examples of closed and open subsets of $\mathbb{R}$, I have nowhere seen an example of a space where compactness is anything different other than the Bolzano-Weierstrass Property, or the Heine-Borel property. Many posts over the web tell us that this definition is applicable to the most general of topological spaces. Can someone provide me an example? What is the motivation behind this definition, or in other words, how do we justify that this is the correct definition of compactness? Just in the scope of real and complex analysis, why do we have to work with such a complicated definition?

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  • $\begingroup$ I think this is a result this definition just turning out to be most useful extension of compactness beyond Euclidean spaces. After all, even the very definition of open sets in topology is just some extension of properties held by (say) open sets in $\mathbb{R}$ induced by metrics. I've seen some nice explanations about this particular point (about definitions of open sets) in a math overflow post, but I can't seem to find it $\endgroup$ – user160738 Mar 9 '17 at 16:28
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    $\begingroup$ As far as I understand it, the concept emerged with Heine's proof in the 1870s that a real continuous function on a closed and bounded interval of $\mathbb{R}$ is in fact uniformly continuous. The open-cover definition falls right out of this argument: I strongly urge you to look at this proof; selecting a finite subcover is precisely what is needed to get a minimum from an infimum. Borel then took up this idea of coverings in the 1890s. The realization was that the extraction of finite subcovers was very useful to make certain types of arguments work. $\endgroup$ – symplectomorphic Mar 9 '17 at 16:35
  • $\begingroup$ To answer a few of your many subquestions: the "Heine Borel" definition of compactness no longer applies in infinitely many dimensions, which is important in functional analysis and PDEs. In every metric space, however, the sequence/subsequence definition is equivalent. Some general (non-metric) topological spaces: Banach spaces under any of various "weak topologies"and affine space/rings under the Zariski topology. $\endgroup$ – Omnomnomnom Mar 9 '17 at 16:44
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    $\begingroup$ After doing some Googling I've discovered something I didn't know: Heine's published proof in 1870/1872 that a continuous function on a closed bounded real interval is uniformly continuous in fact repeated an earlier proof of Dirichlet from 1854 almost word for word. See this. $\endgroup$ – symplectomorphic Mar 9 '17 at 16:46
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    $\begingroup$ @symplectomorphic Very interesting comments. $\endgroup$ – Jean Marie Mar 9 '17 at 16:52
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For every question, there is separate argument why the general notion of compactness is better. I guess that the final definition is a result of gradual improvements. It is hard to explain it simply. So let's go through some examples.

Heine-Borel property

A simple statement of the property is "closed and bounded".

Compactness should be a property of space, not a property of a set within a space. This is the reason why the definition "closed and bounded" fails. For instance the set of rational numbers in interval $[0, 1]$, or the interval $(0,1)$ are also topological (also metric) spaces. They are bounded and although they are not closed in $\mathbb R$, they are closed in themselves.

Instead of closed sets, we have to use another notion of metric spaces, completeness.

And we are not done, yet. There is still a problem with the boundedness. You can define another metric on the real line, $\rho(x, y) = \min(1, |x-y|)$. This is still a correctly defined metric on $\mathbb R$ and it produces the same topology, so if $\mathbb R$ is not compact, the modified space should still not be compact.

But the space happens to be bounded. So instead of bounded space, we need another notion, totally bounded space.

Now, it finally suffices. Actually, a metric space is compact if and only if it is complete and totally bounded. But

  1. the standard definition is much simpler,
  2. the notion of completeness and total bounded space requires the metric, while the standard definition uses just topology -- open sets. So the standard definition is more general.

Weierstrass Property

i.e. Every sequence has a convergent subsequence.

Sequences are bad in general topology. The reason is that you are used to a fact that whenever there is a point $x$ in a closure of a set $S$, then there is a sequence from $S$ converging to $x$ in metric spaces. This is not true in general. For example, you can take the space $\mathbb R$ and add a point $\infty$ (I do not mean usual $+\infty$) such that $\infty$ is in a closure of a set $S$ if and only if $S$ is uncountable. Then there is no sequence of real numbers converging to $\infty$ although $\infty$ is in the closure of real numbers.

There are more examples why sequences are bad but they are often more complicated and require set theory. An example of space which is not compact but every sequence has a convergent subsequence is omega 1

On the other hand, there is an equivalent definition of compactness similar to the Weierstrass Property: For every infinite set $S$ of size $\kappa$, there is a $\kappa$-accumulation point $x$. It means that every open set containing $x$ contains $\kappa$ elements of $S$.

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