coverCover2002-01-04 18:34:382009-01-01 13:25:15Definitionopen coversubcoverrefinementfinite covercountable coveruncountable coveropen subcoveropen refinementcover refinementtopologyset theory% this is the default PlanetMath preamble. as your knowledge
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{\bf Definition} (\cite{kelley}, pp. 49)
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 $U_i$
is a subset of $X$, and
$$ Y \subset \bigcup_{i\in I} U_i.$$
The collection of sets can be arbitrary, that is, $I$ can be
finite, countable, or uncountable. The cover is correspondingly called a
\textbf{finite cover}, \textbf{countable cover}, or \textbf{uncountable cover}.
%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 $$X\subseteq\bigcup_{i\in I}U_i$$
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{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 $X$.
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}.
Open subcovers and open refinements are defined similarly.
{\bf Examples}
\begin{enumerate}
\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 A topology for a set is a cover of that set.
\end{enumerate}
\begin{thebibliography}{9}
\bibitem{kelley} J.L. Kelley, \emph{General Topology},
D. van Nostrand Company, Inc., 1955.
\end{thebibliography}