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\lhead{\tiny Math in Moscow, HSE, Spring 2013} 
\lfoot{\tiny Issued \firstdate} 
\cfoot{-- \thepage \ -- } \rfoot{\tiny Test assignment \#1,\ \ \sc\version}
\rhead{{\tiny  Misha Verbitsky}}


\begin{document}

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\listok{1}{Test assignment \#1}

{\scriptsize
{\bf Rules:} Please solve this in class, before 21:30
February 04, 2013, and give me the written solutions.
The score for the test is computed using the formula
$s= 2p- (\max(p-4,0))$ (rounded down). Results will be
announced at \url{http://bogomolov-lab.ru/KURSY/GEOM-2013/}
}

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\exercise
Find a smooth map $\R\arrow S^3$ with dense image,
or prove that it does not exist.
\ez

\exercise
Find a smooth map $[0,1]\arrow S^3$ with dense image,
or prove that it does not exist.
\ez

\exercise
Let $Z\subset M$ be a dense subset in a manifold,
and $U\supset Z$ an open subset of $M$ containing $Z$.
Show that $U=M$, or find a counterexample.
\ez

\exercise
Let $M$ be a countable, connected metric space. 
Show that $M$ is never infinite.
\ez

\exercise[2 points]
Let $M$ be a compact Hausdorff space,
$R$ -- ring of continuous functions on
$M$ with values in $\Z/2\Z$, and $I\subset R$
a maximal ideal. Prove that there exists
a point $x\in M$ such that all functions
$f\in I$ vanish at $x$.
\ez

\exercise
A continuous map $f:\; X \arrow Y$ of topological spaces
is called {\bf proper} if a preimage of any compact set 
is compact, {\bf closed} if an image of any closed set 
is closed, and {\bf open} if an image of any
open set is open.
Find  an example of a continuous map $f:\; X \arrow Y$
 of Hausdorff topological
spaces which is
\enum 
\ite open, not proper, not closed
\ite[2 points] closed, not proper, not open
\ite[2 points] proper, not open, not closed
\ee
or show that it does not exist.
\ez

\exercise
Let $M:= \R^2 \backslash \Q^2$. Prove that $M$
is connected.
\ez

\exercise
Let $X = \R P^n$, and $Y= (S^1)^m$.
Show that any continous map $f:\; X \arrow Y$ 
is homotopic to a trivial one.
\ez

\exercise
Let $Z\subset \R^n$ be a countable set.
Construct a function $\mu:\; \R^n \arrow \R$
which is continuous at $x\notin Z$ and discontinuous at $Z$.
\ez

\exercise
Let $f_i:\; [0,1]\arrow [0,1]$ be a sequence
of continuous functions, and $f(z):= \lim_i f_i(z)$.
Prove that $f$ is continuous, and find a counterexample.
\ez

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