Mathematical Structures: T1-spaces

# T1-spaces

http://mathcs.chapman.edu/structuresold/files/T1-spaces.pdf
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\begin{document}
\textbf{\Large T1-spaces}

\abbreviation{Top$_1$}

\begin{definition}
A \emph{$T_1$-space} is a \href{Topological_spaces.pdf}{topological spaces} $\mathbf{X}=\langle X,\Omega(\mathbf{X})\rangle$ such that

for every pair of distinct points in the space, there is a pair of open sets containing each point but not the other:  $x,y\in X\implies\exists U,V\in\Omega(\mathbf{X})[x\in U\setminus V\mbox{ and }y\in V\setminus U]$
\end{definition}

\begin{morphisms}
Let $\mathbf{X}$ and $\mathbf{Y}$ be $T_1$-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{definition}
A \emph{$T_1$-space} is a \href{Topological_spaces.pdf}{topological spaces} $\mathbf{X}=\langle X,\Omega(\mathbf{X})\rangle$ such that all

singleton subsets are closed:  $X\setminus\{x\}\in\Omega(\mathbf{X})$
\end{definition}

\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/}
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\begin{subclasses}\

\href{Hausdorff_spaces.pdf}{Hausdorff spaces}

\end{subclasses}

\begin{superclasses}\

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

\end{superclasses}

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

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\bibitem{Ln19xx}

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