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Open Covers and Compact Sets

  1. May 11, 2014 #1
    1. The problem statement, all variables and given/known data
    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.

    2. Relevant 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.
     
  2. jcsd
  3. May 11, 2014 #2
    Rather than completely reword the definitions, I will maybe build up to them in a (perhaps) more intuitive way.

    Throughout, let ##(X,d)## be a metric space, and let ##S\subset X##.

    1a) Let ##\mathcal{S}## be a finite collection of subsets of ##X## indexed by the set ##\{1,2,...,n\}##; i.e. ##\mathcal{S}=\{S_1,S_2,...,S_n\}##, where ##S_i\subset X## for ##i=1,...,n##. We say that ##\mathcal{S}## is a finite cover of ##S## if ##S\subset\cup_{i=1}^nS_i##. We say that this cover is open if ##S_i## is open in ##X## for all ##i=1,...,n##.

    1b) Let ##\mathcal{S}## be a countable collection of subsets of ##X## indexed by the natural numbers; i.e. ##\mathcal{S}=\{S_1,S_2,...,S_n,...\}##, where ##S_i\subset X## for ##i\in\mathbb{N}##. We say that ##\mathcal{S}## is a countable cover of ##S## if ##S\subset\cup_{i\in\mathbb{N}}S_i##. We say that this cover is open if ##S_i## is open in ##X## for all ##i\in\mathbb{N}##.

    1) Let ##\mathcal{S}## be an arbitrary collection of subsets of ##X## indexed by the set ##\mathcal{A}##; i.e. ##\mathcal{S}=\{S_\alpha\}_{\alpha\in\mathcal{A}}##, where ##S_\alpha\subset X## for ##\alpha\in\mathcal{A}##. We say that ##\mathcal{S}## is a cover of ##S## if ##S\subset\cup_{\alpha\in\mathcal{A}}S_\alpha##. We say that this cover is open if ##S_\alpha## is open in ##X## for all ##\alpha\in\mathcal{A}##.

    The visual representation of what a cover is in my head is like stitching the sets in the cover into a blanket (the union) and seeing if our set is "covered" by that blanket (i.e. contained in that union). the words finite, countable, and open are just extra descriptors telling us what kind of cover we are looking at.

    2) Then a family ##\mathcal{S}'## is a subcover of ##S## (relative to the cover ##\mathcal{S}##) if (i) ##\mathcal{S}'\subset\mathcal{S}## (i.e. every subset of ##X## in the family ##\mathcal{S}'## is also in the family ##\mathcal{S}##) and (ii) ##\mathcal{S}'## is a cover of ##S##.

    Basically a subcover is just a smaller (in general, though not necessarily) chunk of the original blanket that still gets the blanketing job done.

    3) The definition of compactness given (sometimes referred to as "covering compactness") says that a set ##S## is compact if whenever we have an arbitrary open cover ##\mathcal{S}## of ##S##, then we can find an finite open subcover ##\mathcal{S}'## (i.e. an subcover that is a finite open cover).
     
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