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\lhead{\large \textbf{Math 461}}
\chead{\Large \textbf{Homework 6}}
\rhead{\large \textbf{DUE 10/15/2023}}
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{\Large\underline{\textbf{Exercises:}}} (\S $17$)



\begin{enumerate}

\item Let $A \subset X$ where $X$ is a topological space.  Prove that
\begin{enumerate}
\item $A = A^\circ$ if and only if $A$ is open in $X$, and that
\item $A = \overline{A}$ if and only if $A$ is closed in $X$.
\end{enumerate}
	
\item Let $X$ be a topological space with subset $S\subset X$. Recall that $\overline{S}$ denotes the closure of $S$ and $S^\circ$ denotes the interior of $S$. We will also denote by $S^c:=X\setminus S$ the complement of $S$.
	\begin{enumerate}
	
	\item Show that $\overline{S}=((S^c)^\circ)^c$ for all $S\subset X$.
	
	\item Show that $S^\circ = (\overline{S^c})^c$ for all $S\subset X$.
	
	\end{enumerate}
	
\item Let $X$ be a topological space and let $A,B\subset X$ be subsets.
	\begin{enumerate}
	\item Show that $A\subset B$ implies $\overline{A}\subset \overline{B}$ and $A^\circ\subset B^\circ$.
	
	\item For $A,B\subset X$, show that $\overline{A\cup B} = \overline{A}\cup\overline{B}$.
	
	\item For $A,B\subset X$, show that $(A\cap B)^\circ = A^\circ \cap B^\circ$.
	
	\item Let $\mathbb{R}$ have the standard topology. Find examples of subsets $A,B\subset \mathbb{R}$ such that $\overline{A\cap B}\neq \overline{A}\cap \overline{B}$ and $(A\cup B)^\circ \neq A^\circ \cup B^\circ$.
	\end{enumerate}
	
\item Let $X$ be a topological space. We say a subset $S\subset X$ is \textbf{dense} in $X$ if for every $x\in X$ and every neighborhood $U$ of $x$ one has $U\cap S\neq\emptyset$. Show the following are equivalent:
	\begin{enumerate}
	\item[(i)] $S$ is dense in $X$.
	
	\item[(ii)] $(S^c)^\circ = \emptyset$.
	
	\item[(iii)] $\overline{S}=X$.
	\end{enumerate}
	
\item If $A \subset X$, we define the {\bf boundary} of $A$ by ${\rm Bd} A := \overline{A} \cap \overline{X \setminus A}$.

	\begin{enumerate}

	\item[(i)] Show that $A^\circ$ and ${\rm Bd} A$ are disjoint.
	\item[(ii)] Show that $\overline{A} = A^\circ \cup {\rm Bd} A$.
	\item[(iii)] Show that ${\rm Bd} A = \overline{A} \setminus A^\circ$ also always holds.
	\item[(iv)] Show that ${\rm Bd} A = \emptyset$ if and only if $A$ is both open and closed.
	\item[(v)] Show that $A$ is open if and only if ${\rm Bd} A = \overline{A} \setminus A$.
	
	\end{enumerate}
	
	
\end{enumerate}




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