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\title{ Math 3210-4,  HW II \\ due Monday Sep 24}
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\noindent 
1) A complex number $z$ is called algebraic if there exists integers $a_0, a_1, \ldots , a_n$, such that  $a_n z^n + \cdots + a_1 z + a_0=0$. Prove that the set of algebraic 
numbers is countable. 

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\noindent 
2) Prove that the following two $(X,d)$ are metric spaces: 
\begin{itemize} 
\item $X=\mathbb R^2$ and $d((x_1,x_2), (y_1,y_2))= \max (|x_1-y_1|, |x_2-y_2|)$. 
\item $X=\mathbb Z$ and $d(x,x)=0$ or $d(x,y)=\frac{1}{2^n}$, if $x\neq y$, where $2^n$ is the largest power of $2$ dividing $x-y$.
\end{itemize} 

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3) Let $(X,d)$ be a metric space. The ``closed'' ball centered at $x$ and of radius $r>0$ is the set of $y\in X$ such that $d(x,y)\leq r$. Prove that the complement of 
the closed ball is an open set in $X$. 

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4) Let $(X,d)$ be a metric space.  Let $E\subset X$. A point $x\in E$ is interior if there exists $\epsilon >0$ such that $B(x,\epsilon) \subset E$.  Let $E^{\circ}$ be the set of 
all interior points of $E$. Prove that $E^{\circ}$ is an open set. If $O \subset E$ is an open set, prove that $O \subset E^{\circ}$. 


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\noindent 
5)  Let $K\subset \mathbb R$ be a set consisting of $0$ and all $1/n$, $n=1,2,3,\ldots$. Prove that $K$ is compact directly using the definition, i.e. every open cover has a 
finite subcover. 

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\noindent 
6) Prove that convex sets in $\mathbb R^2$ are connected, in the sense of definition 2.45. (The proof is very short if you do it the right way.) 

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