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\noindent
\parbox{1.5in}{\bf Math 4710/5710} 
\hfill {\Large \bf  In the Midst-term Exam --- Take Home} \hfill
\parbox{1.5in}{\bf \hfill November $14^{\mathrm{th}}$, 2014}

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\force \hfill {\large
{\bf Due:} Tuesday, November $18^{\mathrm{th}}$, 2014 at 5pm. 
} \hfill \force

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\noindent {\large\bf Name:} \underline{\hskip 3.0 truein} \hfill {\bf Be sure to show your work!}

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The following are True/Possible/False questions. If the statement is always true, state ``{\sc True}'' and then prove the statement. If the statement is never true, state ``{\sc False}'' and then prove the statement cannot ever hold. If the state is sometimes true and sometimes false, state ``{\sc Possible}'' and then give an example of it holding and an example of it failing to hold.

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Each problem is worth {\bf 10 points}. Undergrads should complete {\bf 5} out of {\bf 7} (you may do them all for potential extra credit). Grad students must complete all of these.

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\begin{enumerate}
\item Let $\mathcal{T}$ be a topology for the set $\mathbb{Z} = \{ \dots, -2,-1,0,1,2,\dots \}$.

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Is it {\sc True}, {\sc Possible}, or {\sc False} \quad that $|\mathcal{T}|=5$? 

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\item Let $X$ be a metrizable topological space.

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Is it {\sc True}, {\sc Possible}, or {\sc False} \quad that $X$ has a non-metrizable subspace? 

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\item $X$ is a path connected topological space.

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Is it {\sc True}, {\sc Possible}, or {\sc False} \quad that $X$ is still path connected when given a coarser topology? 
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\item Let $X$ be a non-empty simply ordered set. Give $X$ the order topology. Let $a,b \in X$ with $a<b$.

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Is it {\sc True}, {\sc Possible}, or {\sc False} \quad that $\overline{(a,b)}=[a,b]$?

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\item Let $X$ be a finite topological space. Let $f:X \to Y$ be continuous.

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Is it {\sc True}, {\sc Possible}, or {\sc False} \quad that $f(X)$ is compact?

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\item Let $f:\mathbb{R} \to \mathbb{R}_\ell$ be a continuous map. 

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Is it {\sc True}, {\sc Possible}, or {\sc False} \quad that $f(0)=0$ and $f(1)=1$? 

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\item Let $A$ be a countable closed subset of $X=[0,1] \times [0,1]$ (the unit square in $\mathbb{R}^2$). Let $f:\mathbb{R} \to A$ be continuous.

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Is it {\sc True}, {\sc Possible}, or {\sc False} \quad that there is an open interval $I=(a,b) \subseteq \mathbb{R}$ such that $f$ restricted to $I$ is constant? 

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\end{enumerate}

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