\documentclass[12pt,
               a4paper,
               article,
               oneside,
               oldfontcommands,
               UKenglish]{memoir}


\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage[scaled]{beramono}
\usepackage[final]{microtype}
\usepackage{amssymb}
\usepackage{mathtools}
\usepackage{amsthm}
\usepackage{thmtools}
\usepackage{babel}
\usepackage{csquotes}
\usepackage[nodayofweek]{datetime}
\usepackage{listings}
\lstset{basicstyle = \ttfamily}
\usepackage{textcomp}
\usepackage{siunitx}
\usepackage{xcolor}
\usepackage{graphicx}
\usepackage[colorlinks, allcolors = uiolink]{hyperref}


\renewcommand*{\chaptitlefont}{\Large\bfseries\sffamily\raggedright}
\setsecheadstyle{\large\bfseries\sffamily\raggedright}
\setsubsecheadstyle{\large\bfseries\sffamily\raggedright}
\setsubsubsecheadstyle{\normalsize\bfseries\sffamily\raggedright}
\setparaheadstyle{\normalsize\bfseries\sffamily\raggedright}
\setsubparaheadstyle{\normalsize\bfseries\sffamily\raggedright}
\setbeforesubsubsecskip{2ex}
\setaftersubsubsecskip{1ex}


\pretolerance = 2000
\tolerance    = 6000
\hbadness     = 6000


\renewcommand{\sfdefault}{phv}
\definecolor{uiolink}{HTML}{0B5A9D}


\declaretheorem[style = definition]{problem}
\declaretheorem[style = definition, sibling = problem]{exercise}

\usepackage[all]{xy}
\usepackage{mathrsfs}

\newcommand{\bN}{\mathbb{N}}
\newcommand{\bR}{\mathbb{R}}
\renewcommand{\:}{\colon}
\newcommand{\sB}{\mathscr{B}}
\newcommand{\sT}{\mathscr{T}}
\newcommand{\longto}{\longrightarrow}
\DeclareMathOperator{\cof}{cof}
\DeclareMathOperator{\disc}{disc}
\DeclareMathOperator{\prd}{prod}
\DeclareMathOperator{\triv}{triv}


\begin{document}


\begin{titlingpage}
    \vspace*{-8em}
    \setlength{\parskip}{1.5ex plus 0.5ex minus 0.2ex}
    \setlength{\parindent}{0pt}

    \begin{flushright}
        \small\today
    \end{flushright}

    \begin{center}
        \vspace{2em}
        {
            \bfseries\sffamily\huge
            MAT3500/MAT4500
            %% Alternatively:
            %% COURSEXXXX \textthreequartersemdash\ COURSE NAME
        }
        \vskip0.2ex
        \Large
        Mandatory assignment 1 of 1
        \vskip1ex
    \end{center}

    \subsubsection*{Submission deadline}
    Friday 18\textsuperscript{th} October \the\year, 14:30 in Canvas
    (\href{https://canvas.uio.no}{\underline{canvas.uio.no}}).

    \subsubsection*{Instructions}
    Note that you have \textbf{one attempt} to pass the assignment. This means that there are no second attempts. 
    
    For MAT4500 students, the mandatory assignment must be written with a typesetting software for writing mathematics (e.g.~LaTeX).
    MAT3500 students can choose between scanning handwritten notes or typing the solution directly on a computer (for instance with \LaTeX).
    The assignment must be submitted as a single PDF file.
    Scanned pages must be clearly legible.
    The submission must contain your name, course and assignment number.

    It is expected that you give a clear presentation with all necessary explanations.
    Remember to include all relevant plots and figures. All aids, including collaboration, are allowed, but the submission must be written by you and reflect your understanding of the subject.
    If we doubt that you have understood the content you have handed in, we may request that you give an oral account.

    \subsubsection*{Application for postponed delivery}
    If you need to apply for a postponement of the submission deadline due to illness or other reasons, you have to contact the Student Administration at the Department of Mathematics (e-mail: \href{mailto:studieinfo@math.uio.no}{studieinfo@math.uio.no}) no later than the same day as the deadline.

    All mandatory assignments in this course must be approved in the same semester, before you are allowed to take the final examination.

    \subsubsection*{Complete guidelines about delivery of mandatory assignments:}
    \unskip
    \begin{center}
        \unskip
        \href{http://www.uio.no/english/studies/admin/compulsory-activities/mn-math-mandatory.html}
        {
            \resizebox{\textwidth}{!}
            {\underline{uio.no/english/studies/admin/compulsory-activities/mn-math-mandatory.html}}
        }
        \vspace{1ex}
        \vfill
        GOOD LUCK!
    \end{center}
\end{titlingpage}

\noindent
The assignment consists of 12~parts, each worth 10~points.  Each answer
must be justified.  A total score of \SI{40}{\percent}, or 48~points,
is sufficient to pass.

\medskip

Let $\bN = \{1, 2, 3, \dots\}$ denote the (countable) set of all natural
numbers.  Let $\bR$ denote the (uncountable) set of real numbers.
Let $\bR$ have the usual topology, associated to the metric $d(p, q)
= |q-p|$.

Let $(X, \sT)$ be a topological space, and $p \in X$ a point.
A \emph{neighborhood basis at~$p$} is a collection $\sB_p$ of
neighborhoods of~$p$ in~$X$ such that for each neighborhood $V$ of $p$
in~$X$ there exists a $U \in \sB_p$ with $U \subset V$.  A neighborhood
basis at~$p$ is \emph{countable} if the collection $\sB_p$ is countable,
in which case it can be indexed as $\sB_p = \{U_n\}_{n \in \bN}$.
It will then be said to be \emph{sequential} if $U_{n+1} \subset U_n$
for each $n \in \bN$.

\begin{problem}
Show that for each metric space~$(X, d)$, with underlying topological
space~$(X, \sT_d)$, and any $p \in X$, the collection $\{B_d(p, 1/n)\}_{n
\in \bN}$ is a sequential neighborhood basis at~$p$.
\end{problem}

\begin{problem}
Suppose that $\{U_n\}_{n \in \bN}$ is a countable neighborhood basis
at~$p$.  Show how to construct a collection $\{V_n\}_{n \in \bN}$ that
is a sequential neighborhood basis at~$p$.
\end{problem}

\begin{problem}
Let $\{U_n\}_{n \in \bN}$ and $\{V_n\}_{n \in \bN}$ be sequential
neighborhood bases at~$p$.  Show how to construct a sequential
neighborhood basis at~$p$, denoted $\{W_n\}_{n \in \bN}$, with the
property that $W_n \subset U_n$ and $W_n \subset V_n$ for each~$n
\in \bN$.
\end{problem}

\begin{problem}
Let $\{U_n\}_{n \in \bN}$ and $\{W_n\}_{n \in \bN}$ be sequential
neighborhood bases for~$X$ at~$p$.  Show that there exists a strictly
increasing function $\phi \: \bN \to \bN$ such that $U_{\phi(n)} \subset W_n$
for each~$n \in \bN$.
\end{problem}

Let $C$ be a countable set, and let $\bR^C = \prod_{c \in C} \bR$
be the set of functions $f \: C \to \bR$, equipped with the product
topology~$\sT^C$.  A subbasis for this topology is given by the collection
$$
\{ \pi_c^{-1}((a,b)) \mid c \in C, a < b \in \bR\}
$$
with elements
$$
\pi_c^{-1}((a,b)) = \{f \: C \to \bR \mid a < f(c) < b \} \,.
$$
A basis for~$\sT^C$ is given by the finite intersections of subbasis
elements.

\begin{problem}
Let $0 \: C \to \bR$ denote the zero function.  Explain why each
neighborhood $V$ of $0$ in $(\bR^C, \sT^C)$ contains a basis element
$$
\pi_{c_1}^{-1}((a_1, b_1)) \cap \dots \cap \pi_{c_t}^{-1}((a_t, b_t))
$$
with $c_s \in C$ and $a_s < 0 < b_s$ for each $1 \le s \le t$.
\end{problem}

\begin{problem}
For each finite subset $F \subset C$ and $m \in \bN$ let
$$
U(F, 1/m) := \{ f \: C \to \bR \: \text{$|f(c)| < 1/m$
for each $c \in F$} \} \,.
$$
Show that the collection
$$
\sB_0 := \{ U(F, 1/m) \mid \text{$F \subset C$ is finite, $m \in \bN$} \}
$$
is a neighborhood basis of $0 \in \bR^C$.
\end{problem}

\begin{problem}
Show that the collection
$$
\{F \mid \text{$F \subset C$ is finite} \}
$$
is countable.
\end{problem}

\begin{problem}
Is $\sB_0$ a countable neighborhood basis of~$0$ in~$\bR^C$?
\end{problem}

Let $D$ be an uncountable set, and let $\bR^D = \prod_{d \in D} \bR$
be the set of functions $g \: D \to \bR$, equipped with the product
topology~$\sT^D$.  Suppose~$(*)$ for Problems 9, 10 and~11 that
$\{U_n\}_{n \in \bN}$ is a countable neighborhood basis of the zero
function~$0 \: D \to \bR$ in $(\bR^D, \sT^D)$.

\begin{problem}
Show that for each $n \in \bN$ there is a finite subset $G_n \subset D$
and a natural number $m_n \in \bN$ such that the basis element
$$
V(G_n, 1/m_n) := \{g \: D \to \bR \:
	\text{$|g(d)| < 1/m_n$ for each $d \in G_n$}\}
$$
for $\sT^D$ is contained in $U_n$.
\end{problem}

\begin{problem}
Show that $\bigcup_{n \in \bN} G_n$ is a proper subset of~$D$, so that
there exists an $e \in D$ with $e \notin G_n$ for each~$n \in \bN$.
\end{problem}

\begin{problem}
Prove that the neighborhood
$$
V(\{e\}, 1) := \{g \: D \to \bR \: |g(e)| < 1\}
$$
of $0$ in~$\bR^D$ does not contain~$U_n$ for any $n \in \bN$.
\end{problem}

\begin{problem}
Does there exist a countable neighborhood basis~$\{U_n\}_{n \in \bN}$
of~$0$ in $(\bR^D, \sT^D)$, as supposed at~$(*)$?
\end{problem}

% ((ETC: Consider linear topologies on a real vector space, generated
% by a neighborhood basis of~$\vec 0$.))

\center{--- END ---}





\end{document}
