Topology

HW7: MA 34100. DUE 3/8.

1. Let X be a metric space. Show that (a) ∅, and all singleton sets (i.e. sets of the form {x} for x ∈ X) are connected; (b) what can be said about sets consisting of two distinct elements? (c) If X = R, show that the complement of a bounded open interval is not connected; show that for a ∈ R, (a,∞)c is connected.

2. (a) Prove that every connected metric space with at least two points is uncountable; (b) are closures and interiors of connected sets always connected? (Given a set A, the interior of A is the set of all interior points of A and denoted by Ao.)

3. Determine the convergence or divergence of: ∞∑ n=3

1

n log(n) log(log(n)) ;

∞∑ n=3

1

n log(n)(log(log(n)))π

4. (a) Suppose {sk}k∈N is a complex sequence. Then for n ∈ N define its arithmetic means by

Sn = s1 + . . . + sn

n ,

and prove that if sk → s as k →∞, then lim Sn = s; (b) construct a sequence {sk} which does not converge, while its arithmetic means converges.

5. If {En}n∈N is a sequence of non-empty bounded sets in a complete metric space X such that En+1 ⊂ En, show that if lim diamEn = 0, then ∩∞n=1En contains exactly one point.

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