0

I know I know I'm brute forcing it but it's only because I'm still brute learning Latex for some undergrad Homework. Any help as to why part (b) sections (i) - (iii) are not appearing would be severely appreciated. The list designations are manual because when I try building it using the \begin{

\documentclass[10pt,letterpaper]{article}
\usepackage[letterpaper,margin=0.75in]{geometry}
\usepackage[utf8]{inputenc}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}


\title{Ryan Flynn HW 1}
\author{Ryan Flynn}

\begin{document}

\section*{Exercises for Section 1.1}

%%%%%%%%%%%%%%%%%%%%%%%%%% PROBLEM 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{2}
On the three-point set X = \{a,b,c\}, the trivial topology has two open sets and the discrete topology has eight open sets.  For each $n=3,\dots,7$, either find a topology on $X$ consisting of $n$ open sets or prove that no such topology exists. 


\subsection*{Solution}
The case $n=2$ only contains the \textbb{trivial topology}.

\begin{enumerate}
    \item[(a)] For $n=3$, the collection $\{\varnothing$, X, \{x\}\}, x \in X  \subseteq X$ constitutes a topology on X (e.g. $\mathcal{T} = \{\varnothing,X,\{a\}\}:
        \begin{enumerate}
            \item[(i)] \: Set containment of $\varnothing$ and $X$ \checkmark
            \item[(ii)] \: Closure under all finite intersections:  \checkmark
                    \begin{itemize}
                        \item $\varnothing \cap S =  \varnothing, S \in \mathcal{T}$.
                        \item $X \cap \{x\}=\{x\},\{x\}\in X $
                    \end{itemize}
            \item[(iii)] \: Closure under all unions in \mathcal{T}: \checkmark
                \begin{itemize}
                    \item $\varnothing \cup S = S, S \in \mathcal{T}$
                    \item $X \cup \{x\} = X, \{x\} \in \mathcal{T}$
                \end{itemize}
        \end{enumerate}
    \item[(b)] For n = 4, the collection $\{\varnothing$, X, \{a,b\}\,\{c\}\}, x \in X  \subseteq X$ constitutes a topology on X (e.g. $\mathcal{T} = \{\varnothing,X,\{a\}\}:
        \begin{enumerate}
        \item[(i)] \: Set containment of $\varnothing$ and $X$ \checkmark
        \item[(ii)] \: Closure under all finite intersections:  \checkmark
            \begin{itemize}
                \item WLOG, we note similar intersections above, as well as $\{a,b\} \cap \{c\} = \varnothing and $X \cap \{a,b\} = \{a,b\}.
            \end{itemize}
        \item[(iii)] \: Closure under all unions in \mathcal{T}: \checkmark
            \begin{itemize}

            \end{itemize}
    \end{enumerate}

\end{enumerate}



%%%%%%%%%%%%%%%%%%%%%%%%%% PROBLEM 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{6}
Define a topology on $\mathbb{R}$ (by listing the open sets within it) that contains the open sets (0,2) and (1,3) and that contains as few open sets as possible.

\subsection*{Solution}

%%%%%%%%%%%%%%%%%%%%%%%%%% PROBLEM 3 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{7}
Let $X$ be a set and assume $p \in X$.  Show that the collection $\mathcal T$, consisting of $\varnothing$, $X$, and all subsets of $X$ containing $p$, is a topology on $X$.  This topology is called the \textbf{particular point topology} on $X$, and we denote it by $PPX_p$


\subsection*{Solution}
%%%%%%%%%%%%%%%%%%%%%%%%%% PROBLEM 3 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{8}
Let $X$ be a set and assume $p \in X$.  Show that the collection $\mathcal T$, consisting of $\varnothing$, $X$, and all subsets of $X$that exclude $p$, is a topology on $X$.  This topology is called the \textbf{excluded point topology} on $X$, and we denote it by $EPPX_p$.

\subsection*{Solution}
%%%%%%%%%%%%%%%%%%%%%%%%%% PROBLEM 4 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{9}
Let $\mathcal{T}$ consist of $\varnothing$, $\mathbb{R}$, and all intervals $(-\infty, p )$ for $p \in \mathbb{R}$.  Prove that $\mathcal{T}$ is a topology on $\mathbb{R}$.

\subsection*{Solution}


%%%%%%%%%%%%%%%%%%%%%%%%%% PROBLEM 5 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section*{Exercises for Section 1.2}

\subsection*{11}
Determine which of the following collections of subsets in $\mathbb{R}$ are bases:

\begin{enumerate}[label=(\alph*)]
    \item $\mathcal{C}_1 = \{(n,n+2)\subseteq \mathbb{R} | n \in \mathbb{Z}\}$
    \item $\mathcal{C}_2 = \{[a,b]\subseteq \mathbb{R} \:  | \: a < b\}$
    \item $\mathcal{C}_3 = \{[a,b] \subseteq \mathbb{R} \: \ \: a \leq b\}$
    \item $\mathcal{C}_4 = \{(-x,x)\subseteq \mathbb{R} \: | \: x \in \mathbb{R}\}$
    \item $\mathcal{C}_5 = \{(a,b) \cup \{b+1\}\subseteq \mathbb{R} \: | \: a < b\}$
\end{enumerate}

\subsection*{Solution}
%%%%%%%%%%%%%%%%%%%%%%%%%% PROBLEM 6 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{13}
Consider the following six topologies defined on $\mathbb{R}$: the trivial topology, the discrete topology, the finite complement topology, the standard topology, the lower limit topology, and the upper limit topology.  Show how they compare to each other (finer, strictly finer, coarser strictly coarser, noncomparable) and justify your claim.

\subsection*{Solution}
%%%%%%%%%%%%%%%%%%%%%%%%%% PROBLEM 7 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{14}
Let $\mathcal{B}$ be the collection of subsets of $\mathbb{Z}$ used in defining the digital line Example 1.10.  Show that $\mathcal{B}$ is a basis for a topology on $\mathbb{Z}$.


\subsection*{Solution}
%%%%%%%%%%%%%%%%%%%%%%%%%% PROBLEM 8 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{15}
An \textbb{arithmetic progression} in $\mathbb{Z}$ is a set
\begin{equation*}
    A_{a,b} = \{\dots,a-2b,a-b,a,a+b,a+2b,\dots\}
\end{equation*}
with $a, b \in \mathbb{Z}$ and $b \neq 0$.  Prove that the collection of arithmetic progressions
\begin{equation*}
    \mathcal{A}=\{A_{a,b} \: | \: a,b \in \mathbb{Z} \text{ and } b \neq 0\}
\end{equation*}
is a basis for a topology on $\mathbb{Z}$.  The resulting topology is called the \textbb{arithmetic progression topology} on $\mathbb{Z}$.

\subsection*{Solution}
%%%%%%%%%%%%%%%%%%%%%%%%%% PROBLEM 9 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{16}
\textbb{Prove Theorem 1.12:} On the plane $\mathbb{R}^2$, let
\begin{equation}
    \mathcal{B} = \{(a,b) \times (c,d) \subseteq \mathbb{R}^2 \: | \: a < b, c < d\}
\end{equation}
\begin{enumerate}[label=(\alph*)]
    \item Show that $\mathcal{B}$ is a basis for a topology on $\mathbb{R}^2$.
    \item Show that the topology, $\mathcal{T}'$, generated by $\mathcal{B}$ is the standard topology on $\mathbb{R}^2$. (Hint: if $\mathcal{T}$ is the standard topology, show that $\mathcal{T} \subseteq \mathcal{T}'$ and $\mathcal{T}' \subseteq \mathcal{T}$).
\end{enumerate}

\subsection*{Solution}

\end{document}
  • Why are you numbering manually instead enumerate letting do it? – Ulrike Fischer Aug 28 at 6:56
  • How do I make my code snippet compilable? – mathematicmango Aug 28 at 6:59
  • By making it a complete LaTeX document, starting with \documentclass{...} up to \end{document}. See also tex.meta.stackexchange.com/questions/228/… – siracusa Aug 28 at 7:14
  • I get a huge number of errors from your code. Fix them. – egreg Aug 28 at 8:04
  • What is the \textbb command? – Bernard Aug 28 at 8:16
3

I get a huge number of errors!

  • You have to add \usepackage{enumitem}

  • Math should always be properly segregated; for instance, \mathcal{T} must appear between $ symbols.

  • The command \textbb is undefined; perhaps you want \textbf.

  • Distinct formulas should be set separately:

    $\varnothing \cap S =  \varnothing, S \in \mathcal{T}$.
    

    should be

    $\varnothing \cap S =  \varnothing$, $S \in \mathcal{T}$.
    
  • Instead of | in the set builder notation, use \mid

Here's an edited version, please compare it carefully with yours.

\documentclass[10pt,letterpaper]{article}
\usepackage[letterpaper,margin=0.75in]{geometry}
\usepackage[utf8]{inputenc}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{enumitem}

\title{Ryan Flynn HW 1}
\author{Ryan Flynn}

\begin{document}

\section*{Exercises for Section 1.1}

% PROBLEM 1 %%%

\subsection*{2}
On the three-point set $X = \{a,b,c\}$, the trivial topology has two open sets 
and the discrete topology has eight open sets.  For each $n=3,\dots,7$, either 
find a topology on $X$ consisting of $n$ open sets or prove that no such topology 
exists. 


\subsection*{Solution}
The case $n=2$ only contains the \textbf{trivial topology}.

\begin{enumerate}[label=(\alph*)]
    \item For $n=3$, the collection $\{\varnothing, X, \{x\}\}$, 
          $x \in X  \subseteq X$ constitutes a topology on $X$ 
          (e.g. $\mathcal{T} = \{\varnothing,X,\{a\}\}$:
        \begin{enumerate}[label=(\roman*)]
            \item Set containment of $\varnothing$ and $X$ \checkmark
            \item Closure under all finite intersections:  \checkmark
                    \begin{itemize}
                        \item $\varnothing \cap S =  \varnothing$, $S \in \mathcal{T}$.
                        \item $X \cap \{x\}=\{x\},\{x\}\in X$
                    \end{itemize}
            \item Closure under all unions in $\mathcal{T}$: \checkmark
                \begin{itemize}
                    \item $\varnothing \cup S = S$, $S \in \mathcal{T}$
                    \item $X \cup \{x\} = X$, $\{x\} \in \mathcal{T}$
                \end{itemize}
        \end{enumerate}
    \item For $n = 4$, the collection $\{\varnothing, X, \{a,b\}\,\{c\}\}$, 
          $x \in X  \subseteq X$ constitutes a topology on $X$ 
          (e.g. $\mathcal{T} = \{\varnothing,X,\{a\}\}$:
        \begin{enumerate}[label=(\roman*)]
        \item Set containment of $\varnothing$ and $X$ \checkmark
        \item Closure under all finite intersections:  \checkmark
            \begin{itemize}
                \item WLOG, we note similar intersections above, as well as 
                      $\{a,b\} \cap \{c\} = \varnothing$ and $X \cap \{a,b\} = \{a,b\}$.
            \end{itemize}
        \item Closure under all unions in $\mathcal{T}$: \checkmark
    \end{enumerate}

\end{enumerate}



% PROBLEM 2 %%%

\subsection*{6}
Define a topology on $\mathbb{R}$ (by listing the open sets within it) that 
contains the open sets $(0,2)$ and $(1,3)$ and that contains as few open sets as possible.

\subsection*{Solution}

% PROBLEM 3 %%%

\subsection*{7}
Let $X$ be a set and assume $p \in X$.  Show that the collection $\mathcal T$, 
consisting of $\varnothing$, $X$, and all subsets of $X$ containing $p$, is a 
topology on $X$.  This topology is called the \textbf{particular point topology} 
on $X$, and we denote it by $PPX_p$


\subsection*{Solution}
% PROBLEM 3 %%%

\subsection*{8}
Let $X$ be a set and assume $p \in X$.  Show that the collection $\mathcal T$, 
consisting of $\varnothing$, $X$, and all subsets of $X$that exclude $p$, is a 
topology on $X$.  This topology is called the \textbf{excluded point topology} 
on $X$, and we denote it by $EPPX_p$.

\subsection*{Solution}
% PROBLEM 4 %%%

\subsection*{9}
Let $\mathcal{T}$ consist of $\varnothing$, $\mathbb{R}$, and all intervals 
$(-\infty, p )$ for $p \in \mathbb{R}$.  Prove that $\mathcal{T}$ is a 
topology on $\mathbb{R}$.

\subsection*{Solution}


% PROBLEM 5 %%%

\section*{Exercises for Section 1.2}

\subsection*{11}
Determine which of the following collections of subsets in $\mathbb{R}$ are bases:

\begin{enumerate}[label=(\alph*)]
    \item $\mathcal{C}_1 = \{(n,n+2)\subseteq \mathbb{R} \mid n \in \mathbb{Z}\}$
    \item $\mathcal{C}_2 = \{[a,b]\subseteq \mathbb{R} \mid a < b\}$
    \item $\mathcal{C}_3 = \{[a,b] \subseteq \mathbb{R} \mid a \leq b\}$
    \item $\mathcal{C}_4 = \{(-x,x)\subseteq \mathbb{R} \mid x \in \mathbb{R}\}$
    \item $\mathcal{C}_5 = \{(a,b) \cup \{b+1\}\subseteq \mathbb{R} \mid a < b\}$
\end{enumerate}

\subsection*{Solution}
% PROBLEM 6 %%%

\subsection*{13}
Consider the following six topologies defined on $\mathbb{R}$: the trivial topology, 
the discrete topology, the finite complement topology, the standard topology, the 
lower limit topology, and the upper limit topology.  Show how they compare to each 
other (finer, strictly finer, coarser strictly coarser, noncomparable) and justify 
your claim.

\subsection*{Solution}
% PROBLEM 7 %%%

\subsection*{14}
Let $\mathcal{B}$ be the collection of subsets of $\mathbb{Z}$ used in defining 
the digital line Example 1.10.  Show that $\mathcal{B}$ is a basis for a topology 
on $\mathbb{Z}$.


\subsection*{Solution}
% PROBLEM 8 %%%

\subsection*{15}
An \textbf{arithmetic progression} in $\mathbb{Z}$ is a set
\begin{equation*}
    A_{a,b} = \{\dots,a-2b,a-b,a,a+b,a+2b,\dotsc\}
\end{equation*}
with $a, b \in \mathbb{Z}$ and $b \neq 0$.  Prove that the collection of 
arithmetic progressions
\begin{equation*}
    \mathcal{A}=\{A_{a,b} \mid a,b \in \mathbb{Z} \text{ and } b \neq 0\}
\end{equation*}
is a basis for a topology on $\mathbb{Z}$.  The resulting topology is called 
the \textbf{arithmetic progression topology} on $\mathbb{Z}$.

\subsection*{Solution}
% PROBLEM 9 %%%

\subsection*{16}
\textbf{Prove Theorem 1.12:} On the plane $\mathbb{R}^2$, let
\begin{equation}
    \mathcal{B} = \{(a,b) \times (c,d) \subseteq \mathbb{R}^2 \mid a < b, c < d\}
\end{equation}
\begin{enumerate}[label=(\alph*)]
    \item Show that $\mathcal{B}$ is a basis for a topology on $\mathbb{R}^2$.
    \item Show that the topology, $\mathcal{T}'$, generated by $\mathcal{B}$ is 
          the standard topology on $\mathbb{R}^2$. (Hint: if $\mathcal{T}$ is 
          the standard topology, show that $\mathcal{T} \subseteq \mathcal{T}'$ and 
          $\mathcal{T}' \subseteq \mathcal{T}$).
\end{enumerate}

\subsection*{Solution}

\end{document}

enter image description here

1

I supposed \textbb (which does not exist) was meant \textbf. Loading enumitem(for the optional argument of enumerate here and there) and chasing the unpaired $ made the code perfectly compilable. Also, note that \mathcal requires being in mathmode: you cannot use some text\mathcal{T} some more text`.

\documentclass[10pt,letterpaper]{article}
\usepackage[letterpaper,margin=0.75in]{geometry}
\usepackage[utf8]{inputenc}
\usepackage{amsmath}
\usepackage{amsfonts}%% not needed: it is already loaded by amssymb
\usepackage{amssymb}
\usepackage{enumitem}


\title{Ryan Flynn HW 1}
\author{Ryan Flynn}

\begin{document}

\section*{Exercises for Section 1.1}

%%%%%%%%%%%%%%%%%%%%%%%%%% PROBLEM 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{2}
On the three-point set X = \{a,b,c\}, the trivial topology has two open sets and the discrete topology has eight open sets. For each $n=3,\dots,7$, either find a topology on $X$ consisting of $n$ open sets or prove that no such topology exists.


\subsection*{Solution}
The case $n=2$ only contains the \textbf{trivial topology}.%% \textbb ???
\begin{enumerate}
    \item[(a)] For $n=3$, the collection $\{∅, X, \{x\}\}, x ∈ X ⊆ X$ constitutes a topology on X (e.g. $\mathcal{T} = \{∅,X,\{a\}\}$:
        \begin{enumerate}
            \item[(i)] \: Set containment of $∅$ and $X$ \checkmark
            \item[(ii)] \: Closure under all finite intersections: \checkmark
                    \begin{itemize}
                        \item $∅ ∩ S = ∅, S ∈ \mathcal{T}$.
                        \item $X ∩ \{x\}=\{x\},\{x\} ∈ X $
                    \end{itemize}
            \item[(iii)] \: Closure under all unions in $\mathcal{T}$: \checkmark
                \begin{itemize}
                    \item $∅ ∪ S = S, S ∈ \mathcal{T}$
                    \item $X ∪ \{x\} = X, \{x\} ∈ \mathcal{T}$
                \end{itemize}
        \end{enumerate}
    \item[(b)] For n = 4, the collection $\{∅, X, \{a,b\}\,\{c\}\}, x ∈ X ⊆ X$ constitutes a topology on X (e.g. $\mathcal{T} = \{∅,X,\{a\}\}$:
        \begin{enumerate}
        \item[(i)] \: Set containment of $∅$ and $X$ \checkmark
        \item[(ii)] \: Closure under all finite intersections: \checkmark
            \begin{itemize}
                \item WLOG, we note similar intersections above, as well as $\{a,b\} ∩ \{c\} = ∅$ and $X ∩ \{a,b\} = \{a,b\}$.
            \end{itemize}
        \item[(iii)] \: Closure under all unions in $\mathcal{T}$: \checkmark
% \begin{itemize} emptylist???
%
% \end{itemize}
    \end{enumerate}

\end{enumerate}



%%%%%%%%%%%%%%%%%%%%%%%%%% PROBLEM 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{6}
Define a topology on $\mathbb{R}$ (by listing the open sets within it) that contains the open sets (0,2) and (1,3) and that contains as few open sets as possible.

\subsection*{Solution}

%%%%%%%%%%%%%%%%%%%%%%%%%% PROBLEM 3 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{7}
Let $X$ be a set and assume $p ∈ X$. Show that the collection $\mathcal T$, consisting of $∅$, $X$, and all subsets of $X$ containing $p$, is a topology on $X$. This topology is called the \textbf{particular point topology} on $X$, and we denote it by $PPX_p$


\subsection*{Solution}
%%%%%%%%%%%%%%%%%%%%%%%%%% PROBLEM 3 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{8}
Let $X$ be a set and assume $p ∈ X$. Show that the collection $\mathcal T$, consisting of $∅$, $X$, and all subsets of $X$that exclude $p$, is a topology on $X$. This topology is called the \textbf{excluded point topology} on $X$, and we denote it by $EPPX_p$.

\subsection*{Solution}
%%%%%%%%%%%%%%%%%%%%%%%%%% PROBLEM 4 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{9}
Let $\mathcal{T}$ consist of $∅$, $\mathbb{R}$, and all intervals $(-∞, p )$ for $p ∈ \mathbb{R}$. Prove that $\mathcal{T}$ is a topology on $\mathbb{R}$.

\subsection*{Solution}


%%%%%%%%%%%%%%%%%%%%%%%%%% PROBLEM 5 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section*{Exercises for Section 1.2}

\subsection*{11}
Determine which of the following collections of subsets in $\mathbb{R}$ are bases:

\begin{enumerate}[label=(\alph*)]
    \item $\mathcal{C}₁ = \{(n,n+2) ⊆ \mathbb{R} | n ∈ \mathbb{Z}\}$
    \item $\mathcal{C}₂ = \{[a,b] ⊆ \mathbb{R} \: | \: a < b\}$
    \item $\mathcal{C}₃ = \{[a,b] ⊆ \mathbb{R} \: \ \: a \leq b\}$
    \item $\mathcal{C}₄ = \{(-x,x) ⊆ \mathbb{R} \: | \: x ∈ \mathbb{R}\}$
    \item $\mathcal{C}₅ = \{(a,b) ∪ \{b+1\} ⊆ \mathbb{R} \: | \: a < b\}$
\end{enumerate}

\subsection*{Solution}
%%%%%%%%%%%%%%%%%%%%%%%%%% PROBLEM 6 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{13}
Consider the following six topologies defined on $\mathbb{R}$: the trivial topology, the discrete topology, the finite complement topology, the standard topology, the lower limit topology, and the upper limit topology. Show how they compare to each other (finer, strictly finer, coarser strictly coarser, noncomparable) and justify your claim.

\subsection*{Solution}
%%%%%%%%%%%%%%%%%%%%%%%%%% PROBLEM 7 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{14}
Let $\mathcal{B}$ be the collection of subsets of $\mathbb{Z}$ used in defining the digital line Example 1.10. Show that $\mathcal{B}$ is a basis for a topology on $\mathbb{Z}$.


\subsection*{Solution}
%%%%%%%%%%%%%%%%%%%%%%%%%% PROBLEM 8 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{15}
An \textbf{arithmetic progression} in $\mathbb{Z}$ is a set %%\textbb ???
\begin{equation*}
    A_{a,b} = \{\dots,a-2b,a-b,a,a+b,a+2b, ... \}
\end{equation*}
with $a, b ∈ \mathbb{Z}$ and $b ≠ 0$. Prove that the collection of arithmetic progressions
\begin{equation*}
    \mathcal{A}=\{A_{a,b} \: | \: a,b ∈ \mathbb{Z} \text{ and } b ≠ 0\}
\end{equation*}
is a basis for a topology on $\mathbb{Z}$. The resulting topology is called the \textbf{arithmetic progression topology} on $\mathbb{Z}$. %\textbb ???

\subsection*{Solution}
%%%%%%%%%%%%%%%%%%%%%%%%%% PROBLEM 9 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{16}
\textbf{Prove Theorem 1.12:} On the plane $\mathbb{R}²$, let %% \textbb ???
\begin{equation}
    \mathcal{B} = \{(a,b) × (c,d) ⊆ \mathbb{R}² \: | \: a < b, c < d\}
\end{equation}
\begin{enumerate}[label=(\alph*)]
    \item Show that $\mathcal{B}$ is a basis for a topology on $\mathbb{R}²$.
    \item Show that the topology, $\mathcal{T}'$, generated by $\mathcal{B}$ is the standard topology on $\mathbb{R}²$. (Hint: if $\mathcal{T}$ is the standard topology, show that $\mathcal{T} ⊆ \mathcal{T}'$ and $\mathcal{T}' ⊆ \mathcal{T}$).
\end{enumerate}

\subsection*{Solution}

\end{document} 

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