\documentclass[11pt]{scrartcl}
\usepackage[headheight=1pt, includeheadfoot, headsep=.7cm, top=1.5cm, bottom=1.3cm, left=2cm, right=2cm]{geometry}
\usepackage{mathtools}
\usepackage{amsmath,amsthm, amscd, amssymb, amsfonts}
\usepackage[english]{babel}
\usepackage{fancyhdr}
\usepackage{enumitem}\setlist{itemsep=0.5em} 
\usepackage{multicol}
\usepackage{multienum}
\usepackage{euler}
\usepackage[OT1]{eulervm}
\renewcommand{\rmdefault}{pplx}
\usepackage{mathabx}




%% header, footer definitions
\pagestyle{fancy}
\fancyhead[L]{Math 441}
\fancyhead[C]{Homework 5} 
\fancyhead[R]{Due on July 31} 


%% environments
% theorem style
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{defn}[theorem]{Definition}

% definition style
\theoremstyle{definition}
\newtheorem{problem}{Problem}

% custom solution environment
\newenvironment{solution}{\noindent\textbf{Solution.}}{\qed}

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\newcommand*{\R}{\mathbb{R}}               % real numbers
\newcommand*{\C}{\mathbb{C}}               % complex numbers
\newcommand*{\Q}{\mathbb{Q}}               % rational numbers
\newcommand*{\Z}{\mathbb{Z}}               % integers
\newcommand*{\N}{\mathbb{N}}               % natural numbers
\newcommand*{\abs}[1]{\lvert #1\rvert}     % absolute values, cardinality


% operators
\DeclareMathOperator*{\E}{\mathbb{E}}          % expectation
\DeclareMathOperator*{\Var}{\mathbf{Var}}      % variance
\DeclareMathOperator*{\Span}{span}             % span
\DeclareMathOperator{\tr}{tr}                  % trace
\DeclareMathOperator{\rank}{rank}              % rank
\DeclareMathOperator{\supp}{supp}              % support
\DeclareMathOperator{\diam}{diam}
\DeclareMathOperator{\sgn}{sgn}



\begin{document}

%\noindent Your name! \hfill  The date 

%\vskip10pt

\begin{problem}
\begin{enumerate}
	\item Let $X$ be a set endowed with the cofinite topology. Prove that a sequence $(x_n)$ converges to a point $x\in X$ if and only if for every $y\ne x$, the set $\{n\in\N \mid x_n= y\}$ is finite.
	\item Consider $\R$ with the cofinite topology. Towards what points does the sequence $x_n=\frac{1}{n}$ converge?
\end{enumerate}
\end{problem}


\begin{problem}
Assume that $X$ is a topological spaces with subbasis $\mathcal S$, and let $x\in X$. Prove that a sequence  $(x_n)$ converges to $x$ if and only if for every $S\in \mathcal S$ containing $x$, there exists some  $N\in \N$ such that $x_n\in S$ for every $n\geq N$.
\end{problem}

\begin{problem}
Let $\{X_i \mid i \in \Lambda\}$ be a collection of topological spaces, and let $(x_n)$ be a sequence of elements in the Cartesian product $\prod_{i\in \Lambda } X_i$. For each $j\in \Lambda$, let $p_j\colon \prod_{i\in \Lambda } X_i\to X_j$ denote the projection to the $j$-th coordinate.
\begin{enumerate}
	\item Endow $\prod_{i\in \Lambda } X_i$ with the product topology.  Prove that $(x_n)$ converges to a point $x\in \prod_{i\in \Lambda } X_i$  if and only if for each $j\in \Lambda$, the sequence of $j$-th coordinates $p_j(x_n)$ in converges to $p_j(x)$ in $X_j$. 
	\item Is the statement in Part 2 true if we consider the box topology on $\prod_{i\in \Lambda } X_i$?
\end{enumerate}
\end{problem}

\begin{problem}
In this exercise we endow $\R$ and $\R^2$ with the standard topologies, and quotients with the quotient topology. For the following quotient spaces, describe a
homeomorphism to a simpler, known space.
\begin{enumerate}
	\item $\R/\sim$ where $x\sim y$ if and only if $\sgn(x)=\sgn(y)$.\footnote{The sign function $\sgn \colon \R \to \R$ is defined by $\sgn(0)=0$ and $\sgn(x)=\frac{x}{\abs{x}}$ for $x\ne 0$.}
	\item $\R^2/\sim$ where $(x,y)\sim (x',y')$ if and only if $x=x'$.
\end{enumerate}
\end{problem}

\begin{problem}
Let $S^n = \{x \in \R^{n+1}\mid ||x|| = 1\}$ be the unit sphere with the induced topology, where $||x||=\sqrt{x_1^2 + \dots+ x_{n+1}^2}$. Consider the quotient space $\R P^n = S^n/ \sim$ where 
where $x \sim y$ if and only if  $y= x$ or $y=-x$. This space is called real projective space.
\begin{enumerate}
	\item Find a continuous bijection $\R P^1\to S^1$.
	\item Consider the disk $D^2=\{x \in \R^2\mid ||x||\leq 1 \}$ with the induced topology from $\R^2$ (note that $D^2$ contains $S^1$ as the \emph{boundary} points). Consider on $D^2$ the equivalence relation $\sim'$ such that $x\sim' -x$ for every $x\in S^1\subseteq D^2$ (i.e. opposite boundary points of $D^2$ are identified).
	
	Find a continuous bijection $\R P^2\to D^2/\sim'$.
\end{enumerate}
\end{problem}



\end{document}