1617-257/Homework Assignment 2 Solutions

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Solve problems 1ab, 2, 3, 4ab in section 4, but submit only the underlined problems/parts. In addition, solve the following problems, though submit only your solutions of problems A and B:

Problem A. Let (X,d) be a metric space. Prove that the metric itself, regarded as a function d\colon X\times X\to{\mathbb R}, is continuous.

Problem B. Let A be a subset of a metric space (X,d). Show that the distance function to A, defined by d(x,A):=\inf_{y\in A}d(x,y), is a continuous function and that d(x,A)=0 iff x\in\bar{A}.

Problem C. Prove the "Lebesgue number lemma": If {\mathcal U}=\{U_\alpha\} is an open cover of a compact space (X,d), then there exists an \epsilon>0 (called "the Lebesgue number of {\mathcal U}, such that every open ball of radius \epsilon in X is contained in one of the U_\alpha's.

Problem D. The Cantor set C is the set formed from the closed unit interval [0,1] by removing its open middle third (\frac13,\frac23), then removing the open middle thirds of the remaining two pieces (namely then removing (\frac19,\frac29) and (\frac79,\frac89)), then removing the open middle thirds of the remaining 4 pieces, and so on. Prove that C is uncountable, compact and totally disconnected (the last property means "the only non-empty connected subsets of C are single points").

Student Solutions

Student 1