Topological spaces

 
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 \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.
 
 
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 \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
 
 
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