\documentclass[12pt,a4paper]{book}
\usepackage[utf8]{inputenc}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{graphicx}
\author{,}
\title{,}
\date{\today}
\theoremstyle{plain}
\newtheorem{thm}{Theorem}[section]
\newtheorem{cor}{Corollary}
\newtheorem{lem}{Lemma}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\theoremstyle{remark}
\newtheorem{rem}{Remark}
\newcommand{\miu}{\mu}
\newcommand{\Hom}{\operatorname{Hom}}
\newcommand{\im}{\operatorname{im}}
\begin{document}
\maketitle
\tableofcontents
\part{A}
\chapter{B}
\section{C}
\subsection*{D}
\subsubsection{E}
AAAAAAAAAAAAAA. A
Let $A$ be a set. $$\sum x_i=A$$ AAAAA
A run-off run--off Cauchy--Riemann equations ---
aaaaaaa aaaaaaaaaaaaaaa aaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaa$aaaaaaaaaaaaaa$aaaa~a
\newpage we know that xyx is dalaoest
and all the questions are obvious for him
otherwise
\begin{thm}[Cauchy--Riemann equations]
Let $X$ be a set.
\end{thm}
\begin{proof}
\end{proof}
\begin{defn}
Let $X$ be a set.
\end{defn}
\begin{rem}
AAA
\end{rem}
A
\! \, \; \quad \qquad
$\miu$
$\im \Hom(V,W)$
$\displaystyle\sum_a^Aaaa^A\prod$ \[\int\oint\oplus\otimes\bigoplus_{i=1}^\infty A_i\]
\[\cap \bigcup_{i=1}\]
\[\cong \equiv \sim \simeq \approx\]
\[\leftarrow \Leftarrow \implies \Leftrightarrow\]
\[\lim_{n\to\infty}\]
\[\bar{A} \tilde{A} \hat{A} \vec{x} \neq \not= <> \leq \geq\]
$\dfrac{a}{b}+A$
\\
$\to \mapsto$
$\{x\in X\}$ $\ni$ $\not\subseteq$ $a|b a\mid b$ $: \colon$
$\mathbf{A} \mathrm{A} \mathcal{ABCDEFGHIJLMNOPQRSTUVWXYZ}\\ \mathfrak{ABCDEFGHIJLMNOPQRSTUVWXYZ}$
$\mathcal{B}$
$\mathrm{d}x$
\textrm{A} \textbf{A} \textit{Aabc} \textsl{Aabc} \textsc{Roman}
\end{document}