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\centerline{\bf Topology Preliminary Exam }

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\centerline{\bf August 25, 2005}

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 Do as many problems as possible.

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\noindent 1. Suppose that $X$ is metrizable. Prove that $X$ is $2^{\text{nd}}$ countable if and only 
if $X$ contains a countable dense subset.
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\noindent 2. Let ${\bold{R}}^1$ be the real line and let $Z$ be the integers. 
Prove or disprove that ${\bold{R/Z}}$ is compact. 

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\noindent 3. Show that ${\bold{R}}^2$ minus a countable set is path connected. Hint: look at the set of 
lines through a given point.

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\noindent 4.Let $p: E\to B$ be a covering map. Suppose that f and g are continuous functions from $I$, the closed 
unit interval, to $E$ such that 
\begin{enumerate}
\item $pf$ = $pg$ 
\item $f(0) \neq g(0)$.
\end{enumerate}
Prove that for all $t$ in $I$, $f(t)\neq g(t)$ 

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\noindent 5. Let $X$ = $RP^2 \vee S^1$, the one-point union of the projective plane 
and a circle.
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a.Compute the fundamental group of $X$
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b. Find all 2-sheeted and 3-sheeted coverings of $X$.
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c. Find the universal cover of $X$.

 

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\noindent 6. Let $X$ be a topological space

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a. Suppose that $X$ has a finite number of components. Prove that each component is open.

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b. Give an example to show that the conclusion is false if $X$ has an infinite number of components.


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\noindent 7. Prove that a compact Hausdorff space is normal

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\noindent 8. For each natural number $n$ let $X_n = \{0,1\}$  where $\{0,1\}$ is given the discrete 
topology. Let $X$ be the product of the spaces $X_n$ with the product topology

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\noindent Let $A$ be the subset of $X$ with the property that if $a \in A$ and $a = \{a_1,a_2,...\}$ then there exists an 
$n$ such that  $0 = a_n = a_{n+2} = a_{n+4} = ...$ and $ 1 = a_{n+1} = a_{n+3}= a_{n+5} = ...$. What is the 
closure of $A$?

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