\documentclass{amsart}

\usepackage{mathrsfs} % for mathscr

\newcommand{\bN}{\mathbb{N}}
\newcommand{\bR}{\mathbb{R}}
\renewcommand{\:}{\colon} % for $f \: X \to Y$
\newcommand{\sB}{\mathscr{B}}
\newcommand{\sT}{\mathscr{T}}
\newcommand{\longto}{\longrightarrow}
\DeclareMathOperator{\cof}{cof}
\DeclareMathOperator{\disc}{disc}
\DeclareMathOperator{\prd}{prod}
\DeclareMathOperator{\triv}{triv}

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

\begin{document}

Let $\bN = \{1, 2, 3, \dots\}$ be the set of positive integers, $(X,
\sT_X)$ a topological space, $(x_n)_{n \in \bN} = (x_1, x_2, x_3,
\dots)$ a sequence in $X$, and $p$ an element in~$X$.

\begin{problem}
Precisely define the meaning of ``the sequence $(x_n)_{n \in \bN}$
converges to the point~$p$ in the space~$(X, \sT_X)$.''
\end{problem}

\noindent
Hereafter, all answers should be justified.

Let $\bN_\infty = \bN \cup \{\infty\}$.  Consider the sequence $(n)_{n
\in \bN} = (1, 2, 3, \dots)$ in $\bN_\infty$, and the element $\infty
\in \bN_\infty$.

\begin{problem}
Let $\sT_{\triv}$ denote the trivial topology on~$\bN_{\infty}$.
Does $(n)_{n \in \bN}$ converge to $\infty$ in $(\bN_\infty,
\sT_{\triv})$?
\end{problem}

\begin{problem}
Let $\sT_{\disc}$ denote the discrete topology on~$\bN_{\infty}$.
Does $(n)_{n \in \bN}$ converge to $\infty$ in $(\bN_\infty,
\sT_{\disc})$?
\end{problem}

\begin{problem}
Let $\sT_{\cof}$ denote the cofinite (= finite complement) topology
on~$\bN_{\infty}$.  Does $(n)_{n \in \bN}$ converge to $\infty$ in
$(\bN_\infty, \sT_{\cof})$?
\end{problem}

For $k \in \bN$, let us use the notation
$$
[[k, \infty]] := \{k, k+1, k+2, \dots, \infty\}
	= \{n \in \bN \mid n \ge k\} \cup \{\infty\} \,.
$$
Let
$$
\sB_\infty = \bigl\{ \{k\}, [[k, \infty]] \mid k \in \bN \bigr\}
$$
be the collection of subsets of $\bN_\infty$ of the form $\{k\}$ or the form
$[[k, \infty]]$, for $k \in \bN$.

\begin{problem}
Show that $\sB_\infty$ is a basis for a topology on $\bN_\infty$.
\end{problem}

Let $\sT_\infty$ be the topology on $\bN_\infty$ generated by
the basis~$\sB_\infty$, and consider 
$(\bN_\infty, \sT_\infty)$.

\begin{problem}
What are the open subsets $U \in \sT_\infty$ that do not contain~$\infty$?
\end{problem}

\begin{problem}
What are the open subsets $V \in \sT_\infty$ that do contain~$\infty$?
\end{problem}

\begin{problem}
Is $(\bN_\infty, \sT_\infty)$ a Hausdorff space?
\end{problem}

\begin{problem}
Show that $(n)_{n \in \bN}$ converges to $\infty$ in
$(\bN_\infty, \sT_\infty)$.
\end{problem}

\begin{problem}
Show that if $\sT$ is a topology on $\bN_\infty$ such that
$(n)_{n \in \bN}$ converges to $\infty$ in 
$(\bN_\infty, \sT)$, then $\sT \subset \sT_\infty$.
\end{problem}

Let $(X, \sT_X)$ be a space, as above.  Given a sequence $(x_n)_{n \in
\bN}$ in $X$, and a $p \in X$, let the function $f \: \bN_\infty \to X$
be defined by $f(n) = x_n$ for each $n \in \bN$ and $f(\infty) = p$.

\begin{problem}
Show that if $f \: (\bN_\infty, \sT_\infty) \to (X, \sT_X)$ is continuous,
then $(x_n)_{n \in \bN}$ converges to $p$ in $(X, \sT_X)$.
\end{problem}

\begin{problem}
Show that if $(x_n)_{n \in \bN}$ converges to $p$ in $(X, \sT_X)$,
then $f \: (\bN_\infty, \sT_\infty) \to (X, \sT_X)$ is
continuous.
\end{problem}

% (We can summarize~(i) and~(j) as saying that $\sT_\infty$ is the finest
% topology on $\bN_\infty$ for which $(n)_{n \in \bN}$ converges to
% $\infty$.  Moreover, (k) and~(l) say that $(x_n)_{n \in \bN}$ converges
% to~$p$ if and only if $f$ is continuous.)

Let $\bR$ have the usual topology, associated to the metric
$d(x, y) = |y-x|$.

\begin{problem}
Show that $(\bN_\infty, \sT_\infty)$ is homeomorphic to
$A = \{1/n \mid n \in \bN\} \cup \{0\}$ with the subspace
topology from~$\bR$.
\end{problem}

Let $(\bN_\infty \times \bN_\infty, \sT_{\prd})$ be the product of
two copies of $(\bN_\infty, \sT_\infty)$.  Given a function $g \:
\bN_\infty \times \bN_\infty \to X$, let $x_{m,n} = g(m,n)$ for $m, n \in
\bN_\infty$, and consider the sequences $(x_{m,n})_{n \in \bN}$ in~$X$,
one for each $m \in \bN_\infty$, and the sequences $(x_{m,n})_{m \in \bN}$
in~$X$, one for each $n \in \bN_\infty$.

\begin{problem}
Show that if $g \: (\bN_\infty \times \bN_\infty, \sT_{\prd}) \to
(X, \sT)$ is continuous, then $(x_{m,n})_{n \in \bN}$ converges to
$x_{m,\infty}$ in $(X, \sT)$ as $n \to \infty$, for each $m \in \bN_\infty$.
\end{problem}

By symmetry, $(x_{m,n})_{m \in \bN}$ will converge to $x_{\infty,n}$
in $(X, \sT)$ as $m \to \infty$, for each $n \in \bN_\infty$, but you
do not need to prove this.

\begin{problem}
Suppose that $(x_{m,n})_{n \in \bN}$ converges to $x_{m,\infty}$ in
$(X, \sT)$ as $n \to \infty$, for each $m \in \bN_\infty$, and that
$(x_{m,n})_{m \in \bN}$ converges to $x_{\infty,n}$ in $(X, \sT)$ as
$m \to \infty$, for each $n \in \bN_\infty$.  Does it follow that $g
\: (\bN_\infty \times \bN_\infty, \sT_{\prd}) \to (X, \sT)$
is continuous?

Hints: Consider the case $X = \bN_\infty \times \bN_\infty$
with a topology~$\sT$ that contains
$$
W = \{(m,n) \in X \mid \text{$2m \le n \le \infty$
	or $2n \le m \le \infty$}\} \,.
$$
Alternatively, use the function $h(x,y) = 2xy/(x^2+y^2)$, $h(0,0) = 0$.
\end{problem}


\end{document}
