\documentclass{article}
\usepackage{graphicx}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{enumerate}
\usepackage{ifthen}
\usepackage{ulem} % for strikethrough \sout{...} command
\usepackage{color}

\usepackage{wasysym}

\setlength{\unitlength}{0.1in}

\setlength{\oddsidemargin}{-0.25in}
\setlength{\evensidemargin}{-0.25in}
\setlength{\topmargin}{-0.25in}
\setlength{\headheight}{0.0in}
\setlength{\headsep}{0.0in}
\setlength{\topskip}{0.0in}
\setlength{\textheight}{9.15in}
\setlength{\textwidth}{7in}

\newcommand{\LEMMA}{\vspace{0.05in} \noindent \fbox{\large\bf Lemma:}  }
\newcommand{\COR}{\vspace{0.05in} \noindent \fbox{\large\bf Corollary:}  }
\newcommand{\DEF}{\vspace{0.05in} \noindent \fbox{\large\bf Definition:}  }
\newcommand{\THM}{\vspace{0.05in} \noindent \fbox{\large\bf Theorem:}  }
\newcommand{\PROP}{\vspace{0.05in} \noindent \fbox{\large\bf Proposition:}  }
\newcommand{\EX}{\vspace{0.05in} \noindent \fbox{\large\bf Example:}  }
\newcommand{\bproof}{\vspace{0.1in} \noindent {\bf Proof:} }
\newcommand{\eproof}{$\blacksquare$ \vspace{0.05in}}

\newcommand{\bdot}{\, {\scriptstyle{\bullet}} \,}
\newcommand{\B}[1]{{\bf {#1}}}
\newcommand{\la}{\langle}
\newcommand{\ra}{\rangle}

\newcommand{\myspace}{\parbox[c][0.25in]{0.25in}{\hfill}}
\newcommand{\varspace}[2]{\mbox{\parbox[c][#1]{#2}{\hfill}}}
\newcommand{\force}{{\noindent $\mbox{\;}$}}

\newcommand{\diagquotient}[2]{\displaystyle{{#1 \atop \;}\hspace*{-0.1in} \mbox{\put(0,0){\line(2,1){2}}} \hspace*{0.1in} {\; \atop #2}}}

\begin{document}

%\pagestyle{empty}

\noindent
\parbox{1.95in}{\bf Math 4710/5710}
\hfill {\Large \bf Homework \#5} \hfill
\parbox{1.95in}{\bf \hfill Due: Mon., Feb. 24${}^\mathbf{th}$\!, 2025}

%\vspace{0.2in}
%
%\noindent {\bf\large Name:} \quad \underline{\large\qquad This is my Name \qquad}

\vspace{0.05in}

\noindent
Please remember when submitting any work via email or in person to\dots\\
\force \hfill \fbox{\fbox{\bf\large\color{red}PUT YOUR NAME ON YOUR WORK!}} \hfill \force

\vspace{0.2in}

\begin{description}
\item[\#1 A Problem of Intervals:] Let $Y=[-1,1]$ be thought of as a subspace of $\mathbb{R}$ (with the standard topology). For each of the following sets: If open, explain why. If not open, explain why not.

$A=\{x \in \mathbb{R} \;|\; \frac12 < |x| < 1 \}$, 
$B=\{x \in \mathbb{R} \;|\; \frac12 < |x| \leq 1 \}$, 
$C=\{x \in \mathbb{R} \;|\; \frac12 \leq |x| < 1 \}$, 
$D=\{x \in \mathbb{R} \;|\; \frac12 \leq |x| \leq 1 \}$, and 
$E=\{x \in \mathbb{R} \;|\; 0 < |x| < 1 \text{ and } 1/x \not\in \mathbb{Z}_{>0} \}$.

\vspace{0.05in}

{\bf [Grad.]} Answer the same questions for $Y$ thought of as a subspace of (a) $\mathbb{R}_\ell$ (the lower limit topology) and of\\ \force\hfill (b) $\mathbb{R}_K$ (the $K$-topology). \hspace*{0.685in} \force

\item[\#2 Open Your Map:] Let $X$ and $Y$ be topological spaces and $f:X \to Y$. We say $f$ is an {\it open map} if for every open set $U \subseteq X$, we have $f(U)$ is open (in $Y$). 

Next, fix some index $i$ (between 1 and $n$) and recall that $\pi_i:X_1 \times X_2 \times \cdot \times X_n \to X_i$ is the projection map $\pi_i(x_1,\dots,x_n)=x_i$ onto the $i^\mathrm{th}$ coordinate. Show $\pi_i$ is an open map. 

\item[\#3 Lines are Easy - Right?] Consider $Y = \{ x \times y \in \mathbb{R}^2 \;|\; y=1-x \}$ (a line with negative slope).\footnote{I am using the weird notation: $x \times y$ instead of the usual $(x,y)$ since this problem involves intervals. Sorry!} If we think of $Y$ as a subspace of $\mathbb{R}^2$ with the standard topology, $Y$ ``look like'' the reals with the standard topology. [Technically, by ``looks like'' I mean ``homeomorphic'' -- more on this later.] 

Describe the topology on $Y$ if we think of it as a subspace of $\mathbb{R}_\ell \times \mathbb{R}$ (lower limit product standard topology). Then describe the topology on $Y$ if we think of it as a subspace of $\mathbb{R}_\ell \times \mathbb{R}_\ell$ (lower limit product lower limit topology).

\item[\#4 Getting Things In Order:] {}{\bf [Grad.]} Let $\mathbb{R}^2_{\mathrm{lex}}$ denote $\mathbb{R} \times \mathbb{R}$ with the order topology coming from the lexicographic ordering of the plane. Let $\mathbb{R}_d$ denote the real numbers with the discrete topology and give $\mathbb{R}_d \times \mathbb{R}$ the product topology (discrete by standard). Show $\mathbb{R}^2_{\mathrm{lex}}$ and $\mathbb{R}_d \times \mathbb{R}$ have the same topology.

\vspace{0.05in} 

{\it Note:} In class we sketched out that $\mathbb{R}^2_{\mathrm{lex}}$'s topology is strictly finer than the standard topology on $\mathbb{R}^2$. This problem better explains why this is the case.


\end{description}

\end{document}