Editing Compact space (section)

== Historical development ==

In the 19th century, several disparate mathematical properties were understood that would later be seen as consequences of compactness.  On the one hand, [[Bernard Bolzano]] ([[#CITEREFBolzano1817|1817]]) had been aware that any bounded sequence of points (in the line or plane, for instance) has a subsequence that must eventually get arbitrarily close to some other point, called a [[Limit point of a sequence|limit point]]. 
Bolzano's proof relied on the [[method of bisection]]: the sequence was placed into an interval that was then divided into two equal parts, and a part containing infinitely many terms of the sequence was selected. 
The process could then be repeated by dividing the resulting smaller interval into smaller and smaller parts – until it closes down on the desired limit point. The full significance of [[Bolzano–Weierstrass theorem|Bolzano's theorem]], and its method of proof, would not emerge until almost 50 years later when it was rediscovered by [[Karl Weierstrass]].<ref>{{harvnb|Kline|1990|pp=952–953}}; {{harvnb|Boyer|Merzbach|1991|p=561}}</ref>

In the 1880s, it became clear that results similar to the Bolzano–Weierstrass theorem could be formulated for [[function space|spaces of functions]] rather than just numbers or geometrical points. 
The idea of regarding functions as themselves points of a generalized space dates back to the investigations of [[Giulio Ascoli]] and [[Cesare Arzelà]].<ref>{{harvnb|Kline|1990|loc=Chapter 46, §2}}</ref> 
The culmination of their investigations, the [[Arzelà–Ascoli theorem]], was a generalization of the Bolzano–Weierstrass theorem to families of [[continuous function]]s, the precise conclusion of which was that it was possible to extract a [[uniform convergence|uniformly convergent]] sequence of functions from a suitable family of functions. The uniform limit of this sequence then played precisely the same role as Bolzano's "limit point". Towards the beginning of the twentieth century, results similar to that of Arzelà and Ascoli began to accumulate in the area of [[integral equation]]s, as investigated by [[David Hilbert]] and [[Erhard Schmidt]]. 
For a certain class of [[Green's functions]] coming from solutions of integral equations, Schmidt had shown that a property analogous to the Arzelà–Ascoli theorem held in the sense of [[mean convergence]] – or convergence in what would later be dubbed a [[Hilbert space]]. This ultimately led to the notion of a [[compact operator]] as an offshoot of the general notion of a compact space. 
It was [[Maurice René Fréchet|Maurice Fréchet]] who, in [[#CITEREFFréchet1906|1906]], had distilled the essence of the Bolzano–Weierstrass property and coined the term ''compactness'' to refer to this general phenomenon (he used the term already in his 1904 paper<ref>Frechet, M. 1904. {{lang|fr|italic=no|"Generalisation d'un theorem de Weierstrass"}}. {{lang|fr|Analyse Mathematique}}.</ref> which led to the famous 1906 thesis).

However, a different notion of compactness altogether had also slowly emerged at the end of the 19th century from the study of the [[linear continuum|continuum]], which was seen as fundamental for the rigorous formulation of analysis. 
In 1870, [[Eduard Heine]] showed that a [[continuous function]] defined on a closed and bounded interval was in fact [[uniformly continuous]]. In the course of the proof, he made use of a lemma that from any countable cover of the interval by smaller open intervals, it was possible to select a finite number of these that also covered it. 
The significance of this lemma was recognized by [[Émile Borel]] ([[#CITEREFBorel1895|1895]]), and it was generalized to arbitrary collections of intervals by [[Pierre Cousin (mathematician)|Pierre Cousin]] (1895) and [[Henri Lebesgue]] ([[#CITEREFLebesgue1904|1904]]). The [[Heine–Borel theorem]], as the result is now known, is another special property possessed by closed and bounded sets of real numbers.

This property was significant because it allowed for the passage from [[local property|local information]] about a set (such as the continuity of a function) to global information about the set (such as the uniform continuity of a function). This sentiment was expressed by {{harvtxt|Lebesgue|1904}}, who also exploited it in the development of the [[Lebesgue integral|integral now bearing his name]]. Ultimately, the Russian school of [[point-set topology]], under the direction of [[Pavel Alexandrov]] and [[Pavel Urysohn]], formulated Heine–Borel compactness in a way that could be applied to the modern notion of a [[topological space]]. {{harvtxt|Alexandrov|Urysohn|1929}} showed that the earlier version of compactness due to Fréchet, now called (relative) [[sequential compactness]], under appropriate conditions followed from the version of compactness that was formulated in terms of the existence of finite subcovers. It was this notion of compactness that became the dominant one, because it was not only a stronger property, but it could be formulated in a more general setting with a minimum of additional technical machinery, as it relied only on the structure of the open sets in a space.
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One of the main reasons for studying compact spaces is because they are in some ways very similar to [[finite set]]s: there are many results which are easy to show for finite sets, whose proofs carry over with minimal change to compact spaces. 
Here is an example:

* Suppose {{mvar|X}} is a [[Hausdorff space]], and we have a point {{math|''x'' ∈ ''X''}} and a finite subset {{mvar|A}} of {{mvar|X}} not containing {{mvar|x}}. Then we can [[separated sets|separate]] {{mvar|x}} and {{mvar|A}} by [[neighborhood (topology)|neighborhoods]]: for each {{math|''a'' ∈ ''A''}}, let {{math|''U''(''x'')}} and {{math|''V''(''a'')}} be disjoint neighborhoods containing {{mvar|x}} and {{mvar|a}}, respectively. Then the intersection of all the {{math|''U''(''x'')}} and the union of all the {{math|''V''(''a'')}} are the required neighborhoods of {{mvar|x}} and {{mvar|A}}.

Note that if {{mvar|A}} is [[Infinity|infinite]], the proof fails, because the intersection of arbitrarily many neighborhoods of {{mvar|x}} might not be a neighborhood of {{mvar|x}}. 
The proof can be "rescued", however, if {{mvar|A}} is compact: we simply take a finite subcover of the cover {{math|{''V''(''a'') : ''a'' ∈ A}}} of {{mvar|A}}, then intersect the corresponding finitely many {{math|''U''(''x'')}}. 
In this way, we see that in a Hausdorff space, any point can be separated by neighborhoods from any compact set not containing it. 
In fact, repeating the argument shows that any two disjoint compact sets in a Hausdorff space can be separated by neighborhoods&nbsp;– note that this is precisely what we get if we replace "point" (i.e. [[singleton set]]) with "compact set" in the Hausdorff [[separation axiom]]. 
Many of the arguments and results involving compact spaces follow such a pattern.
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