5. Compact SetsJuly 12, 2019LetSbe a subset ofR. IfSis bounded and closed, then it is an important propertycalled: compactness. The definition of “compactness” looks bit strange but it is very useful The definition of compact sets Let S ⊆ R be a subset. 1. A collection F of some open sets in R such that all of open sets in F covers S , namely, S ⊆ ∪ A ∈F A, is called an open cover of S . 2. A subset G of F which also covers S is called a subcover of S . If the number of all elements of G is finite, G is called a finite subcover of S . 3. S is compact if for every open cover F of S , it has a finite subcover of S , namely, there is a finite open set A 1 , A 2 , ..., A n as elements of F such that 1 S ⊆ ∪ n j =1 A j . (1) The historical origin The above definition of compactness appears strange. Let us go back history to find its trace. At some time, central to the theory of analysis was the concept of uniform continuity and a theorem stating that Theorem 0.1 Every continuous function on a closed finite interval is uniformly continuous. 1 In short, S is compact if every open cover of S has a finite subcover. 41
2 T.H. Hildebrandt, The Borel Theorem and Its Generalizations , Bulletin of the AMS, vol. 32 (1926), pp. 423-474. 42
You've reached the end of your free preview.
Want to read all 9 pages?
- Fall '08
- Topology, Metric space, Borel, Bolzano weierstrass theorem, set S