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我知道我在强行使用它，但这只是因为我仍在为一些本科作业而强行学习 Latex。任何关于为什么第 (b) 部分第 (i) - (iii) 节未出现的帮助都将不胜感激。列表指定是手动的，因为当我尝试使用 \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}
答案1
我收到大量错误！
你必须添加
\usepackage{enumitem}数学应该始终被适当地隔离；例如，
\mathcal{T}必须出现在$符号之间。该命令
\textbb未定义；也许您想要\textbf。应分别设置不同的公式：
$\varnothing \cap S = \varnothing, S \in \mathcal{T}$.应该
$\varnothing \cap S = \varnothing$, $S \in \mathcal{T}$.|不要使用集合构建器符号，而是使用\mid
这是编辑后的版本，请仔细与您的版本进行比较。
\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}
答案2
我认为\textbb（不存在）的意思是 \textbf。加载enumitem（对于这里和那里的可选参数enumerate）并追逐不成对的$使得代码完全可编译。另请注意，\mathcal需要处于数学模式：您不能使用some text\mathcal{T} 一些更多文本。
\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}