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| A \emph{filter subbasis} for a set $S$ is a collection of subsets of $S$ which has the finite intersection property. |
A \emph{filter subbasis} for a set $S$ is a collection of subsets of $S$ which has the finite intersection property. |
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A \emph{filter basis} $B$ for a set $S$ is a collection of subsets of $S$ which does not contain the empty set such that, for every $u \in B$ and every $v \in B$, there exists a $w \in B$ such that $w \subset u \cap v$.
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A \emph{filter basis} $B$ for a set $S$ is a collection of subsets of $S$ which does not contain the empty set such that, for every $u \in B$ and every $v \in B$, there exists a $w \in B$ such that $w \in u \cap v$.
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| Given a filter basis $B$ for a set $S$, the set of all supersets of elements of $B$ forms a filter on the set $S$. This filter is known as the filter generated by the basis. |
Given a filter basis $B$ for a set $S$, the set of all supersets of elements of $B$ forms a filter on the set $S$. This filter is known as the filter generated by the basis. |
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| Given a filter subbasis $B$ for a set $S$, the set of all supersets of finite intersections of elements of $B$ is a filter. This filter is known as the filter generated by the subbasis. |
Given a filter subbasis $B$ for a set $S$, the set of all supersets of finite intersections of elements of $B$ is a filter. This filter is known as the filter generated by the subbasis. |
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| Two filter bases are said to be \PMlinkescapetext{equivalent} if they generate the same filter. Likewise, two filter subbases are said to be equivalent if they generate the same filter. |
Two filter bases are said to be \PMlinkescapetext{equivalent} if they generate the same filter. Likewise, two filter subbases are said to be equivalent if they generate the same filter. |
