# 1617-257/Homework Assignment 2

## Doing

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

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

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

Problem D. The Cantor set ${\displaystyle C}$ is the set formed from the closed unit interval ${\displaystyle [0,1]}$ by removing its open middle third ${\displaystyle ({\frac {1}{3}},{\frac {2}{3}})}$, then removing the open middle thirds of the remaining two pieces (namely then removing ${\displaystyle ({\frac {1}{9}},{\frac {2}{9}})}$ and ${\displaystyle ({\frac {7}{9}},{\frac {8}{9}})}$), then removing the open middle thirds of the remaining 4 pieces, and so on. Prove that ${\displaystyle C}$ is uncountable, compact and totally disconnected (the last property means "the only non-empty connected subsets of ${\displaystyle C}$ are single points").