1617-257/Homework Assignment 2 Solutions

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Student Solutions

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 $({\frac {1}{3}},{\frac {2}{3}})$, then removing the open middle thirds of the remaining two pieces (namely then removing $({\frac {1}{9}},{\frac {2}{9}})$ and $({\frac {7}{9}},{\frac {8}{9}})$), 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 1: HW2 Sol

Student 2: