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COMPACT, a tutorial by Scott W. Williams 

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5. OPEN COVERS
Borel proved the following in his 1894 thesis: A countable covering of a closed interval by open intervals has a finite subcover [Hildebrandt 1924].
It turns out that Borel's approach was similar to the approach Heine used to prove in 1872 that a continuous function on a closed interval was uniformly continuous (actually first proved, but unpublished for 60 years, by Dirichlet in 1852).
In 1898, Lebesgue (and apparently someone named Cousins in 1895) removed "countable" from the hypothesis and the generalized theorem, which is now commonly called the HeineBorel theorem, with modern notation, is:
5.1. THEOREM. A subset of
is compact iff it is closed and bounded.
Unfortunately, this notion of "bounded" does not work for metric spaces, and even "metric" might not be present. Vietoris' [Vietoris1921] seems to have first seriously considered abstract compact spaces and he proved 1.4(4) and proved "expansion of disjoint closed sets to disjoint open sets" (see theorem 3.1), but, independently, Alexandrov and Urysohn [Alexandrov and Urysohn1923] first gave it the modern definition (though it was called bicompact for many years):
5.2. DEFINITION. A space X is compact if each covering by open sets contains finitely many open sets which cover.
5.3. EXAMPLES.
1. With the sequence {0}, with {} we see that an open set containing 0 (or ) contains all but finitely many of the points. To see that [0,1] is compact, use the least upper bound property.
There is an important consequence of compactness which at first appeared to be a property of completeness. It is now known as the Baire Category Theorem due to Baire for the reals in 1889 and Hausdorff for complete metric spaces in 1914. It was Cech who saw the earlier results were a consequence for certain subsets of a compact spaces [Cech1937]:
5.4. THEOREM. Suppose {G_{}n: n} is a countable family of open dense sets in a compact space. Then the intersection G_{n} is dense.
Note that a complete metric spaces can be embedded as a such an intersection in a compact space. Using 5.4, Banach gave, in 1931, an elegant proof of theWeierstrass (1852) and Bolzano [1833] result:
5.5. COROLLARY. There is a continuous nowhere differentiable function from the reals to the reals.
page 1: the beginnings 
page 2: a special example 
page3: functional separation 

page4: the universe in a box 
page5: open covers 