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\usepackage{multicol}
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\usepackage{euler}
\usepackage[OT1]{eulervm}
\renewcommand{\rmdefault}{pplx}
\usepackage{mathabx}




%% header, footer definitions
\pagestyle{fancy}
\fancyhead[L]{Math 441}
\fancyhead[C]{Homework 1} 
\fancyhead[R]{Due on June 26} 


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% theorem style
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\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
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\newcommand*{\N}{\mathbb{N}}               % natural numbers
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\begin{document}

%\noindent Your name! \hfill  The date 

%\vskip10pt


\begin{problem}
Let $X$ be a topological space and $A$ a subset of $X$. Define the \emph{interior} of $A$ (denoted $A^{\circ}$) and the \emph{closure} of $A$ (denoted $\overline A$) by 
\begin{align*}
\mathring A&=\{x \in X \mid \text{$A$ is a neighborhood of $x$} \} \\
	\overline A&=\{x \in X \mid \text{$X\setminus A$ is not a neighborhood of $x$} \} 
\end{align*}
Prove that $\mathring A \subseteq A \subseteq \overline A$. 
\end{problem}

%\begin{solution}

%\end{solution}
\begin{problem}
Let $X$ be a topological space and $A$ a subset of $X$. Prove that 
\begin{align*}
	\mathring A&=\bigcup_{U \text{ open, }  U \subseteq A} U, & \overline A&=\bigcap_{C \text{ closed, }  C \supseteq A} C.
\end{align*}
\end{problem}

\begin{problem}[10 points]
Let $X$ be a topological space and $A$ a subset of $X$. 
\begin{enumerate}
\item Prove that $\mathring A$ is the largest open set contained in $A$ and that $\overline A$ is the smallest closed set containing $A$. 
\item\label{prob2part2} Prove that $A$ is open if and only if $A=\mathring A$ if and only if $A$ is a neighborhood of each of its points. 
\item State and prove the closed version of the first biconditional (if and only if) of Part \ref{prob2part2}.
\item For the standard topology on $\R$, compute the interior and the closure of the interval $[0,2)$.
\end{enumerate}
\end{problem}


\begin{problem}
Let $X$ be any set. Prove that the family of subsets 
\[\mathcal T_{\text {cf}}=\{ U \subseteq X \mid X\setminus U \text { is finite } \} \cup \{ \emptyset\}\]
defines a topology on $X$. This is called the \emph{cofinite} topology.
\end{problem}

\begin{problem}
Let $X$ be a topological space and let $A$ be a subset of $X$. Prove that the family of subsets 
\[\mathcal T_{A}=\{ V \subseteq A \mid \text {there exists an open set } U \subseteq X \text { such that } V=U\cap A\} \]
defines a topology on $A$. This is called the \emph{induced} topology.
\end{problem}

\begin{problem}
Munkres 13.3
\end{problem}


\begin{problem}
	Munkres 13.4
\end{problem}

\begin{problem}
	Munkres 13.6
\end{problem}


\begin{problem}
	Munkres 13.8 
\end{problem}

\end{document}