COMPACT, a tutorial

Scott W. Williams

State University of New York in Buffalo

title and preface

page 1:
the beginnings

page 2:
a special example

functional separation


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to prove in 1872 that a continuous function on a closed interval was uniformly continuous (actually first proved, but unpublished 60 years, by Dirichlet in 1852).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.