- #1
analysis001
- 21
- 0
Homework Statement
I'm trying to understand what compact sets are but I am having some trouble because I am having trouble understanding what open covers are. If someone could reword the following definitions to make them more understandable that would be great.
Homework Equations
Definition: Suppose (X, d) is a metric space and S[itex]\subseteq[/itex]X. We will say that the family of sets {S[itex]\alpha[/itex]}[itex]\alpha\in[/itex]A is an open cover of S if all S[itex]\alpha[/itex], [itex]\alpha\in[/itex]A, are open sets and if S[itex]\subseteq\cup_{\alpha\in A}[/itex]S[itex]\alpha[/itex].
Definition: Given an open cover {S[itex]\alpha[/itex]} of S, we will furthermore say that the family {S[itex]\beta[/itex]}[itex]\beta\in[/itex]B, is a subcover of {S[itex]\alpha[/itex]} if B[itex]\subseteq[/itex]A and {S[itex]\beta[/itex]} is an open cover of S.
Definition: Suppose (X, d) is a metric space an S[itex]\subseteq[/itex]X. We will say that S is a compact set if every open cover {S[itex]\alpha[/itex]} of S has a finite subcover.