This chapter introduces the important concept of compact sets. Compact sets defined in any metric (or topological) spaces are a generalizations of closed real intervals known from the real analysis of one variable. The study of compact sets allows a deeper understanding of theorems known from the analysis of continuous functions with one real variable on bounded and closed intervals. The following table compares the results about compact sets with some basic results known from the real analysis of one variable:
|Real Analysis with One Variable
|Definition of Closed Real Intervals
|Compact Subsets of Metric Spaces Are Bounded and Closed.
|Continuous Real Functions on Closed Intervals Take Maximum and Minimum Values within these Intervals
|Continuous Functions Mapping Compact Domains to Real Numbers Take Maximum and Minimum Values on these Domains.
|Continuous Real Functions on Closed Intervals are Bounded
|Continuous Functions Mapping Compact Domains to Real Numbers are Bounded.