5Let us conclude these notes with a discussion of the most disconnected set possible –theCantor set.Example 20.LetCdenote the Cantor set (cf.§.244 in ). Recall thatCmay be describedas the set of alleven ternary numbers; that is,C⊂[0,1]consists of numbersxwhose base-3expansion[x]3has only0s and2s. Letx∈C, and fix>0. Then, by the Archimedeanproperty, there isn∈Nwithn>1. Note that3n> n, and so>13n>0. Now, writexinbase-3,[x]3= 0.x1x2x3. . . xn-1xnxn+1. . .This is just short-hand for the expansionx=x13+x232+x333+· · ·+xn3n+· · ·Now, consider the two numbersx+andx-given byx±=x±13n.Ifxn= 0then[x+]3= 0.x1x2. . . xn-11xn+1. . .; ifxn= 2then[x-]3= 0.x1x2. . . xn-11xn+1. . ..In either case,x±do not have even ternary expansions, and sox±/∈C. A similar argu-ment (though with more work due to carrying, as in “carry the1. . . ”) shows thatx±/∈Cin general. Note also that, by choice ofn,x±∈B(x,).
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Empty set, Metric space, Topological space, General topology, Clopen set