Mathematical Structures: Topological spaces

Topological spaces

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 %%run pdflatex % \documentclass[12pt]{amsart} \usepackage[pdfpagemode=Fullscreen,pdfstartview=FitBH]{hyperref} \parindent=0pt \parskip=5pt \addtolength{\oddsidemargin}{-.5in} \addtolength{\evensidemargin}{-.5in} \addtolength{\textwidth}{1in} \theoremstyle{definition} \newtheorem{definition}{Definition} \newtheorem*{morphisms}{Morphisms} \newtheorem*{basic_results}{Basic Results} \newtheorem*{examples}{Examples} \newtheorem{example}{} \newtheorem*{properties}{Properties} \newtheorem*{finite_members}{Finite Members} \newtheorem*{subclasses}{Subclasses} \newtheorem*{superclasses}{Superclasses} \newcommand{\abbreviation}[1]{\textbf{Abbreviation: #1}} \pagestyle{myheadings}\thispagestyle{myheadings} \markboth{\today}{math.chapman.edu/structures} \begin{document} \textbf{\Large Topological spaces} \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Topological_spaces}{edit} \abbreviation{Top} \begin{definition} A \emph{topological space} is a structure $\mathbf{X}=\langle X,\tau\rangle$, where $\tau=\Omega(\mathbf{X})\subseteq P(X)$ is a collection of subsets of $X$ called the \emph{open sets of} $\mathbf{X}$ such that any union of open sets is open: $\mathcal{U}\subseteq\Omega(\mathbf{X})\implies\bigcup\mathcal{U}\in\Omega(\mathbf{X})$ any finite intersection of open sets is open: $U,V\in\Omega(\mathbf{X})\implies U\cap V\in\Omega(\mathbf{X})$ and $X\in\Omega(\mathbf{X})$ Remark: Note that the union of an empty collection is empty, so $\emptyset\in\Omega(\mathbf{X})$. The set of \emph{closed sets of} $\mathbf{X}$ is $K(\mathbf{X})=\{X-U\mid U\in\Omega(\mathbf{X})\}$. \end{definition} \begin{morphisms} Let $\mathbf{X}$ and $\mathbf{Y}$ be topological spaces. A morphism from $\mathbf{X}$ to $\mathbf{Y}$ is a function $f:X\rightarrow Y$ that is \emph{continuous}: $V\in\Omega(\mathbf{Y})\implies f^{-1}[V]\in\Omega(\mathbf{X})$ \end{morphisms} \begin{basic_results} \end{basic_results} \begin{examples} \begin{example} \end{example} \end{examples} \begin{table}[h] \begin{properties} (\href{http://math.chapman.edu/cgi-bin/structures?Properties}{description}) \begin{tabular}{|ll|}\hline Classtype & second-order\\\hline Amalgamation property & yes\\\hline Strong amalgamation property & yes\\\hline Epimorphisms are surjective & yes\\\hline \end{tabular} \end{properties} \end{table} Remark: The properties given above use an $(\mathcal{E},\mathcal{M})$ factorization system with $\mathcal{E}=$ surjective morphisms and $\mathcal{M}=$ embeddings. \hyperbaseurl{http://math.chapman.edu/structures/files/} \parskip0pt \begin{subclasses}\ \href{T0-spaces.pdf}{T0-spaces} \end{subclasses} \begin{superclasses}\ \href{Sets.pdf}{Sets} \end{superclasses} http://www.wikipedia.org/wiki/Topological_space \begin{thebibliography}{10} \bibitem{Ln19xx} \end{thebibliography} \end{document} %
 %%run pdflatex % \documentclass[12pt]{amsart} \usepackage[pdfpagemode=Fullscreen,pdfstartview=FitBH]{hyperref} \parindent=0pt \parskip=5pt \addtolength{\oddsidemargin}{-.5in} \addtolength{\evensidemargin}{-.5in} \addtolength{\textwidth}{1in} \theoremstyle{definition} \newtheorem{definition}{Definition} \newtheorem*{morphisms}{Morphisms} \newtheorem*{basic_results}{Basic Results} \newtheorem*{examples}{Examples} \newtheorem{example}{} \newtheorem*{properties}{Properties} \newtheorem*{finite_members}{Finite Members} \newtheorem*{subclasses}{Subclasses} \newtheorem*{superclasses}{Superclasses} \newcommand{\abbreviation}[1]{\textbf{Abbreviation: #1}} \pagestyle{myheadings}\thispagestyle{myheadings} \markboth{\today}{math.chapman.edu/structures} \begin{document} \textbf{\Large Topological spaces} \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Topological_spaces}{edit} \abbreviation{Top} \begin{definition} A \emph{topological space} is a structure $\mathbf{X}=\langle X,\tau\rangle$, where $\tau=\Omega(\mathbf{X})\subseteq P(X)$ is a collection of subsets of $X$ called the \emph{open sets of} $\mathbf{X}$ such that any union of open sets is open: $\mathcal{U}\subseteq\Omega(\mathbf{X})\implies\bigcup\mathcal{U}\in\Omega(\mathbf{X})$ any finite intersection of open sets is open: $U,V\in\Omega(\mathbf{X})\implies U\cap V\in\Omega(\mathbf{X})$ and $X\in\Omega(\mathbf{X})$ Remark: Note that the union of an empty collection is empty, so $\emptyset\in\Omega(\mathbf{X})$. The set of \emph{closed sets of} $\mathbf{X}$ is $K(\mathbf{X})=\{X-U\mid U\in\Omega(\mathbf{X})\}$. \end{definition} \begin{morphisms} Let $\mathbf{X}$ and $\mathbf{Y}$ be topological spaces. A morphism from $\mathbf{X}$ to $\mathbf{Y}$ is a function $f:X\rightarrow Y$ that is \emph{continuous}: $V\in\Omega(\mathbf{Y})\implies f^{-1}[V]\in\Omega(\mathbf{X})$ \end{morphisms} \begin{basic_results} \end{basic_results} \begin{examples} \begin{example} \end{example} \end{examples} \begin{table}[h] \begin{properties} (\href{http://math.chapman.edu/cgi-bin/structures?Properties}{description}) \begin{tabular}{|ll|}\hline Classtype & second-order\\\hline Amalgamation property & yes\\\hline Strong amalgamation property & yes\\\hline Epimorphisms are surjective & yes\\\hline \end{tabular} Remark: The properties given above use an $(\mathcal{E},\mathcal{M})$ factorization system with $\mathcal{E}=$ surjective morphisms and $\mathcal{M}=$ embeddings. \end{properties} \end{table} \hyperbaseurl{http://math.chapman.edu/structures/files/} \begin{subclasses}\ \href{T0-spaces.pdf}{T0-spaces} \end{subclasses} \begin{superclasses}\ \href{Sets.pdf}{Sets} \end{superclasses} \url{http://www.wikipedia.org/wiki/Topological_space} \begin{thebibliography}{10} \bibitem{Ln19xx} \end{thebibliography} \end{document} %

http://mathcs.chapman.edu/structuresold/files/Topological_spaces.pdf
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\documentclass[12pt]{amsart}
\usepackage[pdfpagemode=Fullscreen,pdfstartview=FitBH]{hyperref}
\parindent=0pt
\parskip=5pt
\theoremstyle{definition}
\newtheorem{definition}{Definition}
\newtheorem*{morphisms}{Morphisms}
\newtheorem*{basic_results}{Basic Results}
\newtheorem*{examples}{Examples}
\newtheorem{example}{}
\newtheorem*{properties}{Properties}
\newtheorem*{finite_members}{Finite Members}
\newtheorem*{subclasses}{Subclasses}
\newtheorem*{superclasses}{Superclasses}
\newcommand{\abbreviation}[1]{\textbf{Abbreviation: #1}}
\markboth{\today}{math.chapman.edu/structures}

\begin{document}
\textbf{\Large Topological spaces}

\abbreviation{Top}

\begin{definition}
A \emph{topological space} is a structure $\mathbf{X}=\langle X,\tau\rangle$, where $\tau=\Omega(\mathbf{X})\subseteq P(X)$
is a collection of subsets of $X$ called the \emph{open sets of} $\mathbf{X}$ such that

any union of open sets is open:  $\mathcal{U}\subseteq\Omega(\mathbf{X})\implies\bigcup\mathcal{U}\in\Omega(\mathbf{X})$

any finite intersection of open sets is open:  $U,V\in\Omega(\mathbf{X})\implies U\cap V\in\Omega(\mathbf{X})$ and $X\in\Omega(\mathbf{X})$

Remark: Note that the union of an empty collection is empty, so $\emptyset\in\Omega(\mathbf{X})$.

The set of \emph{closed sets of} $\mathbf{X}$ is $K(\mathbf{X})=\{X-U\mid U\in\Omega(\mathbf{X})\}$.

\end{definition}

\begin{morphisms}
Let $\mathbf{X}$ and $\mathbf{Y}$ be topological spaces.
A morphism from $\mathbf{X}$ to $\mathbf{Y}$ is a function $f:X\rightarrow Y$ that is \emph{continuous}:

$V\in\Omega(\mathbf{Y})\implies f^{-1}[V]\in\Omega(\mathbf{X})$
\end{morphisms}

\begin{basic_results}
\end{basic_results}

\begin{examples}
\begin{example}
\end{example}
\end{examples}

\begin{table}[h]
\begin{properties} (\href{http://math.chapman.edu/cgi-bin/structures?Properties}{description})

\begin{tabular}{|ll|}\hline
Classtype & second-order\\\hline
Amalgamation property & yes\\\hline
Strong amalgamation property & yes\\\hline
Epimorphisms are surjective & yes\\\hline
\end{tabular}

Remark:
The properties given above use an $(\mathcal{E},\mathcal{M})$ factorization system with $\mathcal{E}=$ surjective morphisms and
$\mathcal{M}=$ embeddings.
\end{properties}
\end{table}

\hyperbaseurl{http://math.chapman.edu/structures/files/}
\begin{subclasses}\

\href{T0-spaces.pdf}{T0-spaces}

\end{subclasses}

\begin{superclasses}\

\href{Sets.pdf}{Sets}

\end{superclasses}

\url{http://www.wikipedia.org/wiki/Topological_space}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

\end{thebibliography}

\end{document}
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