\documentclass{article}

\title{\textbf{Homework 3}}
%\author{Enrique Trevi\~no}

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\begin{document}

\maketitle
\normalsize
\section{Chapter 2}
\begin{problem}(\textbf{Exercise 3.1.1})\\
Prove that every constant function is continuous (using Theorem 3.1).
\end{problem}
\begin{sol}
\end{sol}

\begin{problem}(\textbf{Exercise 3.1.3})\\
Define the function $f$ over the domain $[-1/\pi, 1/\pi]$ by
$$f(x) = \sin(1/x),\,\, x\ne 0, \,\,\,\, f(0) = 0.$$
Describe $f^{-1}(N_{1/2}(0))$, the inverse image of $N_{1/2}(0)$. Is this inverse image open, closed, or neither?
\end{problem}
\begin{sol}
\end{sol}

\begin{problem}(\textbf{Exercise 3.1.5})\\
For each of the following sets in $\mathbb{R}^2$, state whether the set is open, closed, or neither. Then find the closure, the interior, and the boundary of the set.
\begin{enumerate}
\item $\{(x,y)\,|\, x > 0\}$
\item $\{(x,y)\,|\, x+y > 1 \mbox{ and } y\le 1\}$
\item $\{(x,y)\,|\, x+y \ge 1,\, x > 0, \mbox{ and } y > 1\}$
\item $\{(x,y)\,|\, y = x^2\}$
\item $\{(x,y)\,|\, x = 1 \mbox{ and } y < 2\}$
\item $\{(x,y)\,|\, 1 \le x \le 2,\, y\in\mathbb{N}\}$
\item $\{(x,y)\,|\, 1\le x\le 2 \mbox{ and } y\in\mathbb{Q}\}$
\end{enumerate}
\end{problem}
\begin{sol}
\end{sol}

\begin{problem}(\textbf{Exercise 3.1.7})\\
Prove that every neighborhood of $x$ contains at least one point of $S$ other than $x$ itself if and only if every neighborhood of $x$ contains infinitely many points of $S$.
\end{problem}
\begin{sol}
\end{sol}

%\begin{problem}(\textbf{Exercise 3.1.8})\\
%Prove that if a set is both open and closed, then it is either $\mathbb{R}$ or the empty set.
%\end{problem}
%\begin{sol}
%\end{sol}

\begin{problem}(\textbf{Exercise 3.1.11})\\
Prove that if $S^{C}$, the complement of $S$, is open, then $S^{C}$ cannot contain an accumulation point of $S$.
\end{problem}
\begin{sol}
\end{sol}

\begin{problem}(\textbf{Exercise 3.1.12})\\
Prove that if $x\in S^{C}$ is not an accumulation point of $S$, then there is an $\varepsilon > 0$ for which $N_{\varepsilon}(x)\subseteq S^{C}$.
\end{problem}
\begin{sol}
\end{sol}

\begin{problem}(\textbf{Exercise 3.1.13})\\
Prove that any union of open sets is open.
\end{problem}
\begin{sol}
\end{sol}

\begin{problem}(\textbf{Exercise 3.1.15})\\
Give an example of an infinite intersection of open sets that is not open.
\end{problem}
\begin{sol}
\end{sol}

\begin{problem}(\textbf{Exercise 3.1.17})\\
Prove that any intersection of closed sets is closed.
\end{problem}
\begin{sol}
\end{sol}

\begin{problem}(\textbf{Exercise 3.1.19})\\
Give an example of an infinite union of closed sets that is not closed.
\end{problem}
\begin{sol}
\end{sol}


\end{document} 