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\chead{Topology Comprehensive Exam}
\lhead{2024 Fall}

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{\large\textbf{Topology Comprehensive Exam}}\hfill{Name:$\underline{\hspace{130pt}}$}
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Answer six (6) questions total. On the first page of your work, please write the numbers of the problems that you want graded. On each page please write only on the front side.
$\\$

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Problems & \makebox[.5cm] & \makebox[.5cm] & \makebox[.5cm] & \makebox[.5cm] & \makebox[.5cm] & \makebox[.5cm] & Total\\ \hline
Scores & & & & & & & \\ \hline
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\begin{description}
\item[1.]
\begin{enumerate}[label=\alph*.]
\item Define the \emph{discrete topology} on a set X. 
\item  Define a $T_1$-space.
\item  Assume $X$ is a $T_1$-space, and for any family of open subsets, $\{A_\lambda\}_{\lambda\in \Lambda}$, the intersection $\bigcap_{\lambda\in \Lambda}A_\lambda$ is also open. Prove that the topology of $X$ is the discrete topology
\end{enumerate}

\item[2.]
\begin{enumerate}[label=\alph*.]
\item Give the definition of the \emph{closure} $\overline{A}$ of a subset $A \subseteq X$.
\item Let $X$ and $Y$ be topological spaces and let $A \subseteq X$. Prove that if $f : X \to Y$
is continuous, then $f(\overline{A}) \subseteq \overline{f(A)}$.
\item Let $X$ be a Hausdorff space, and let $A\subseteq X$. Suppose there exists $x \in \overline{A}$ such
that $x \not\in A$. Prove that every open set that contains $x$ must contain infinitely many points of $A$.
\end{enumerate}

\item[3.]
\begin{enumerate}[label=\alph*.]
\item  Let $X$ be a topological space. Give the definition of a dense subset of $X$.
\item Assume $U$ is an open subset of $X$, and $A$ is a dense subset of $X$. Prove that
$U \subseteq \overline{A \cap U}$, where $\overline{A \cap U}$ denotes the closure of $A \cap U$ in $X$.
\end{enumerate}

\item[4.]
\begin{enumerate}[label=\alph*.]
\item Define what it means for $X$ to be \emph{connected} and for $X$ to be \emph{path connected}.
\item Prove that the continuous image of a connected space is connected.
\item Give an example of a space that is connected but not path connected. You do not need to justify your answer.
\end{enumerate}

\item[5.]
\begin{enumerate}[label=\alph*.]
\item Define what it means for a topological space $X$ to be \emph{normal}.
\item Prove that a compact subset of a $T_2$ space is closed.
\item Prove that a compact $T_2$ space is normal.
\end{enumerate}

\item[6.]
If $X$ is an infinite set, then the \emph{cofinite topology} on $X$ consists of the collection of subsets of $X$ whose complement is finite, together with the empty set.
\begin{enumerate}[label=\alph*.]
\item Prove that the cofinite topology on $X$ is indeed a topology.
\item Prove that $\R$ with the cofinite topology is connected.
\item Is $\R$ with the cofinite topology Hausdorff? Justify your answer.
\item  Prove that $\R$ with the cofinite topology is compact.
\end{enumerate}

\newpage
\item[7.]
\begin{enumerate}[label=\alph*.]
\item Let $X$, $Y$, and $Z$ be topological spaces.
Prove that if $f:X\to Y$ is continuous and $g:Y\to Z$ is continuous, then $g\circ f: X\to Z$ is continuous.

\item Let $X= \prod_{\alpha \in A}X_\alpha$ be a product of topological spaces and let $Y$ be a topological space.
Let $\pi_\alpha: X\to X_\alpha$ be the projection map to the factor with index $\alpha$.
Prove that a function $f:Y\to X$ is continuous if and only if $\pi_\alpha\circ f: Y\to X_\alpha$ is continuous for each $\alpha \in A$.
\end{enumerate}

\item[8.]
\begin{enumerate}[label=\alph*.]
\item Prove that if $X$ is a compact space and $Y$ is a Hausdorff space, and if $f :X\to Y$ is continuous, one-one, and onto, then $f$ is a homeomorphism.
\item Prove that  if $f (X,\tau)$ is a compact Hausdorff space and $\tau'$ is (strictly) weaker than $\tau$ then $(X,\tau' )$ is not Hausdorff.

\item Prove that if $(X,\tau)$ is a compact Hausdorff space and $\tau '$ is (strictly) stronger than $\tau$, then $(X,\tau' )$ is not compact.
\end{enumerate}

\item[9.]
For each of the collection of properties below, give an example of a topological space with those properties. You do not need to justify your answers.
\begin{enumerate}[label=\alph*.]
\item $T_1$ and not $T_2$ 
\item $T_2$ and not $T_3$ 
\item $T_3$ and not $T_4$
\item Lindel\"{o}f and not 2nd-countable
\item First countable and not 2nd-countable
\end{enumerate}

\item[10.]
\begin{enumerate}[label=\alph*.]
\item Define \emph{quotient map}.
\item Given an equivalence relation $\sim$ on a topological space $X$, define the quotient topology on the set of equivalence classes $X/\sim$.
\item Let $X = \R^3 - \{\mathbf{0}\}$ with the subspace topology inherited from the product topology on $\R^3$. Define an equivalence relation $\sim$ on X given by $\mathbf{x} \sim \mathbf{y}$ if there exists a positive constant $\lambda$ such that $\mathbf{x} = \lambda\mathbf{y}$. Give the well-known topological space to which $X/ \sim$ with the quotient topology is homeomorphic.
\end{enumerate}

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