Notes

Description: In this note we include definitions from Keremedis [1998a] for forms [1 BL] through [1 BR], [1CB], and [67 G].

Content:

In this note we include definitions from Keremedis [1998a] for forms [1 BL] through [1 BR], [1CB], and [67 G].

In what follows, \((X,T)\) is a topological space and \(P\subset\cal P(X)\).
Definition: A non-empty set \(\cal F\subseteq P- \{\emptyset\}\) is called a \(P\)-filter iff

  1. \(F_1\), \(F_2\in\cal F\) implies \(F_1\cap F_2\in\cal F\).
  2. \(F\in\cal F\), \(F\subset F'\) and \(F'\in P\) implies \(F'\in\cal F\).
A \(T\)-filter is called an open filter. \(\cal F\) is called a filter base if it just satisfies (1.) above.

Definition:  A non-empty set \()\cal I\subseteq P- \{X\}\) is called a \(P\)-ideal iff

  1. \(I_1\), \(I_2\in\cal I\) implies \(I_1\cup I_2\in\cal I\).
  2. \(I\in\cal I\), \(I'\subset I\) and \(I'\in P\) implies \(I'\in\cal I\).
If \(F\) is the set of closed subsets of \(X\), then a \(F\)-ideal is called a closed ideal.  \(\cal I\) is called complete if \(\bigcup C\in\cal I\) for every \(C\subseteq\cal I\).

Definition: A disjoint set \(C\subseteq T-\{\emptyset\}\) is called a cellular family of \(X\).

Definition: A nest of \((X,T)\) is a maximal linearly ordered subset of the partially ordered set \((T-\{\emptyset\}, \subseteq)\).

Definition: A tower \(\cal T\) of \((X,T)\) is a linearly ordered subset of the partially ordered set \((T-\{\emptyset\}, \subseteq)\) such that either the interior of \(\bigcap\cal T\) is empty or the interior of \(\bigcap\cal T\) does not properly contain any non-empty open set.

Definition: If \(\{(X_i,T_i): i\in k\}\) is a family of topological spaces, then the box topology on \(\prod_{i\in k} X_i\) is the topology having a base of all sets of the form \(\prod_{i\in k}O_i\) where \(O_i\in T_i\) for all \(i\in  k\).

Definition: If \(\{(X_i,T_i): i\in k\}\) is a family of disjoint topological spaces, then the free union is the set \(X= \bigcup\{X_i: i\in k\}\) together with the topology \(T\) containing all sets \(O\subseteq X\) such that \(O\cap X_i\in T_i\) for all \(i\in k\).

Howard-Rubin number: 77

Type: Definitions

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