This holds for all connected c containing p thus

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This holds for all connected C containing p , thus their union C p is also a subset of A . But then C p B = C p A B = contradicting that C p B 6 = . 6. Let ( X, d ) be the discrete metric space; i.e., d(x,y)=1 if x 6 = y , d ( x, y ) = 0 if x = y . (a) Can X be connected? Explain. (b) Can X be compact? Explain. Solution. In a discrete space all subsets are both open and closed. Except if X is a singleton set (or empty), X contains at least one open and closed set that is neither empty nor X . Thus X is connected if and only if it is either empty or a singleton set. Concerning compactness, the family {{ x } : x X } is an open covering of X that admits NO proper subcovering; it follows that X will be compact if and only if X is finite. 2
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  • Fall '09
  • Schonbek
  • Topology, Metric space, Topological space

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