is compact iff
it is closed and bounded.
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COMPACT, a tutorial Scott W. Williams State University of New York in Buffalo |
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6. 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to prove in 1872 that a continuous function on a closed interval was uniformly continuous (actually first proved, but unpublished 60 years, by Dirichlet in 1852). The generalized theorem, which is now commonly called the Heine-Borel theorem, with modern notation, is:
THEOREM5. A subset of
is compact iff
it is closed and bounded.
Cantor, Fréchet, Poincaré, Hilbert, Alexandrov, Urysohn, Hausdorff, Birkhoff, Lindelöf, ...
I was invited to present a one hour tutorial on Topology for students attending the fifth annual Conference for African American Researchers in the Mathematical Sciences held June 22-26, 1999 at the University of Michigan - Ann Arbor. This is the lecture.
copy permission each person is allowed one electronic version of this paper and one hard copy printout.
