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

\vspace{0.1in}

\noindent
{\bf Standard assumptions:} $X$ is a topological space. Subsets are given the subspace topology. Products are given the product topology.

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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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\noindent {\bf\large 1. (12 points)} An open question.
\begin{enumerate}[(a)]
\item Using the standard basis for $\mathbb{R}_\ell$ (real numbers with lower limit topology), show $I=(0,1)$ is open.

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\item The interval $J=(0,1]$ is not open in $\mathbb{R}$. Why? 

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\item The interval $J=(0,1]$ is open in $X=(-\infty,1]$. Why?

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\end{enumerate}

\noindent {\bf\large 2. (10 points)} Let $\mathcal{T} = \{ U \subseteq X \;|\; U \mbox{ is finite.} \} \cup \{ X \}$.
\begin{enumerate}[(a)]
\item Explain why $\mathcal{T}$ is not a topology for $X$.

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\item Can we restrict $X$ so that $\mathcal{T}$ is a topology for $X$?

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%\item Suppose that we find the smallest $\mathcal{T}'$ such that $\mathcal{T} \subseteq \mathcal{T}'$ and $(X,\mathcal{T}')$ is a topological space.
%          What can we say about this space?

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\end{enumerate}

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\noindent {\bf\large 3. (9 points)} Let $A \subseteq X$. Suppose that $a_n \in A$ for all $n=1,2,3,\dots$ and $a_n \to a$. Show that $a \in \bar{A}$.

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\noindent {\bf\large 4. (13 points)} Let $\displaystyle f : A \to \prod\limits_{j \in J} X_j$ ($A$ and $X_j$'s are topological spaces). For each $a \in A$, we have $f(a) = (f_j(a))_{j \in J}$ for some functions $f_j:A \to X_j$. 
\begin{enumerate}[(a)]
\item Suppose  $f$ is continuous. Show that $f_j$ is continuous for all $j \in J$.

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\item Suppose that $f_j$ is continuous for all $j \in J$. Show that $f$ is continuous.

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\item Let $f: \mathbb{R} \to \mathbb{R}^\omega$ be defined by $f(t)=(t,t,t,\dots)$. Explain why $f$ is continuous if $\mathbb{R}^\omega$ is given the product topology but not continuous if given the box topology.

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\end{enumerate}

\newpage

\noindent {\bf\large 5. (9 points)} Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces. 
\begin{enumerate}[(a)]
\item Using the language of metric spaces, give a careful definition of what it means for $f:X \to Y$ to be continuous at $x \in X$.

\vspace{1in}

\item Using the language of topological spaces, give a careful definition of what it means for $f:X \to Y$ to be continuous at $x \in X$ (this should work if $X$ and $Y$ are just topological spaces and not necessarily metric spaces).

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\item Briefly explain why these concepts are equivalent.

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\end{enumerate}

\noindent {\bf\large 6. (7 points)} Basic continuity. $X$, $Y$, and $Z$ are topological spaces. Suppose $f:X \to Y$ and $g:Y \to Z$ are continuous. Show $g \circ f$ is continuous.


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\noindent {\bf\large 7. (10 points)} Let $f:X \to Y$ be a continuous map between topological spaces $X$ and $Y$ where $X$ is connected. Show $f(X)$ is connected.


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\noindent {\bf\large 8. (10 points)} Not connected. A statement or a hint?
\begin{enumerate}[(a)]
\item Show $X = [0,1] \cup [2,3]$ is not connected.

\vspace{1in}

%\item Show $R_\ell$ is not connected.

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\item Explain why $\mathbb{R} \not\cong S^1$ (the unit circle).

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\end{enumerate}

\noindent {\bf\large 9. (10 points)} Compact and not compact.
\begin{enumerate}[(a)]
\item Let $X$ have the finite complement topology. Show $X$ is compact.

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\item Show the interval $I=(0,1) \subset \mathbb{R}$ is not compact.

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\end{enumerate}

\noindent {\bf\large 10. (10 points)} Let $A \subset X$ where $X$ is Hausdorff and $A$ is compact. Suppose that $x_0 \not\in A$. Show that there exists open sets $U$ and $V$ such that $U$ and $V$ are disjoint, $x_0 \in U$ and $A \subseteq V$.


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%\noindent {\bf\large 11. (points)} Let $f: X \to Y$ be a continuous bijection. Suppose that $X$ is compact and $Y$ is Hausdorff. Show that $f$ is a homeomorphism.

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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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