| Version current |
Version 14 |
| \PMlinkescapeword{collection} |
\PMlinkescapeword{collection} |
| {\bf Definition} (\cite{kelley}, pp. 49) |
{\bf Definition} (\cite{kelley}, pp. 49) |
| Let $Y$ be a subset of a set $X$. A \textbf{cover} for $Y$ is a collection |
Let $Y$ be a subset of a set $X$. A \textbf{cover} for $Y$ is a collection |
| of sets $\mathcal{U}=\{U_i\}_{i\in I}$ such that each $U_i$ |
of sets $\mathcal{U}=\{U_i\}_{i\in I}$ such that each $U_i$ |
| is a subset of $X$, and |
is a subset of $X$, and |
| $$ Y \subset \bigcup_{i\in I} U_i.$$ |
$$ Y \subset \bigcup_{i\in I} U_i.$$ |
| The collection of sets can be arbitrary, that is, $I$ can be |
The collection of sets can be arbitrary, that is, $I$ can be |
| finite, countable, or uncountable. The cover is correspondingly called a |
finite, countable, or uncountable. The cover is correspondingly called a |
| \textbf{finite cover}, \textbf{countable cover}, or \textbf{uncountable cover}. |
\textbf{finite cover}, \textbf{countable cover}, or \textbf{uncountable cover}. |
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| %Let $X$ be a set and let $\mathbb{P}(X)$ denote the power set of $X$. A %collection $\mathcal{U}=\{ U_i\in\mathbb{P}(X) \colon i\in I\}$ of subsets of $X$ %is said to be a \emph{cover} of X if $$X\subseteq\bigcup_{i\in I}U_i$$ |
%Let $X$ be a set and let $\mathbb{P}(X)$ denote the power set of $X$. A %collection $\mathcal{U}=\{ U_i\in\mathbb{P}(X) \colon i\in I\}$ of subsets of $X$ %is said to be a \emph{cover} of X if $$X\subseteq\bigcup_{i\in I}U_i$$ |
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| A \textbf{subcover} of $\mathcal{U}$ is a subset $\mathcal{U}'\subset\mathcal{U}$ such that $\mathcal{U}'$ is also a cover of $X$. |
A \textbf{subcover} of $\mathcal{U}$ is a subset $\mathcal{U}'\subset\mathcal{U}$ such that $\mathcal{U}'$ is also a cover of $X$. |
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| A \textbf{refinement} $\mathcal{V}$ of $\mathcal{U}$ is a cover of $X$ such that for every $V\in\mathcal{V}$ there is some $U\in\mathcal{U}$ such that $V\subset U$. When $\mathcal{V}$ refines $\mathcal{U}$, it is usually written $\mathcal{V}\preceq \mathcal{U}$. $\preceq$ is a preorder on the set of covers of any topological space $X$. |
A \textbf{refinement} $\mathcal{V}$ of $\mathcal{U}$ is a cover of $X$ such that for every $V\in\mathcal{V}$ there is some $U\in\mathcal{U}$ such that $V\subset U$. When $\mathcal{V}$ refines $\mathcal{U}$, it is usually written $\mathcal{V}\preceq \mathcal{U}$. $\preceq$ is a preorder on the set of covers of any topological space $X$. |
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| If $X$ is a topological space and the members of $\mathcal{U}$ are open sets, |
If $X$ is a topological space and the members of $\mathcal{U}$ are open sets, |
| then $\mathcal{U}$ is said to be an \emph{open cover}. |
then $\mathcal{U}$ is said to be an \emph{open cover}. |
| Open subcovers and open refinements are defined similarly. |
Open subcovers and open refinements are defined similarly. |
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| {\bf Examples} |
{\bf Examples} |
| \begin{enumerate} |
\begin{enumerate} |
| \item If $X$ is a set, then $\{X\}$ is a cover of $X$. |
\item If $X$ is a set, then $\{X\}$ is a cover of $X$. |
| \item The power set of a set $X$ is a cover of $X$. |
\item The power set of a set $X$ is a cover of $X$. |
| \item A topology for a set is a cover of that set. |
\item A topology for a set is a cover of that set. |
| \end{enumerate} |
\end{enumerate} |
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| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
| \bibitem{kelley} J.L. Kelley, \emph{General Topology}, |
\bibitem{kelley} J.L. Kelley, \emph{General Topology}, |
| D. van Nostrand Company, Inc., 1955. |
D. van Nostrand Company, Inc., 1955. |
| \end{thebibliography} |
\end{thebibliography} |
