Description:
The following definitions and results for forms [8 M],
[8 N],
[43 Q], [126 B] through
[126 F] are from Keremedis [2000a].
Content:
The following definitions and results for forms [8 M],
[8 N],
[43 Q], [126 B] through
[126 F] are from Keremedis [2000a].
Let \((X,T)\) be a topological space. \(B\subseteq T-\{\emptyset\}\) is said to be a \(\pi\)-base for \(X\) if every non-empty element of \(T\)
includes an element of \(B\).
Let \((P,\le)\) be a partially ordered set (poset).
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A subset \(D\subseteq P\) is set to be dense if every element of \(P\) has a lower bound in \(D\). \(D\) is said to be open
if it is open in the order topology.
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An element \(a\in P\) is said to be an atom if every pair of lower bounds of \(a\) are compatible (have a common lower bound).
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If \(P\) has no atoms it is called atomless. A pairwise incompatible subset of \(P\) is called an antichain.
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Let \(D=\{D_i: i\in\omega\}\) be a denumerable family of dense open sets in \(P\). \(C=\{C_i: i\in\omega\}\), \(\bigcup C=\omega\)
is called a matrix for \(D\) if the following three condition are satisfied:
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\(\forall i\in\omega\) \(C_i\) is a countable antichain in \(P\).
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\(\forall i\in\omega\), \(\forall j>i\), \(\forall c\in C_i\) there exist infinitely many \(t\in C_j\) such that \(t\le c\).
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\(\forall i\in\omega\), \(\forall j\in\omega\),all but finitely many element of \(C_j\) belong to \(D_i\).
Let \(A=\{A_i: i\in I\}\) be a family of non-empty sets.
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\(A\) has the strong finite intersection property, sfip if for each finite \(Q\subseteq A\), \(|\bigcap Q|\ge\omega\).
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An infinite subset \(S\) of \(A\) is a pseudo-intersection if \(|S-A_i|<\omega\) for all \(i\in I\).
The following statements are considered:
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\(MC_{\omega}(\aleph_0,\infty)\): For every denumerable family \(A\) of pairwise disjoint non-empty sets, there exists a set \(C\) such that for all
\(a\in A\), \(0<|C\cap a|\le\omega\).
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\(pseudo(\omega)\) ([126 F]) : Every denumerable family of sets having the sfip also has a
pseudo-intersection.
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\(DP\): If \((X,T)\) is a topological space having a countable \(\pi\)-base,then for every family \(D=\{D_i: i\in\omega\}\) of dense open sets of
\(X\), there is a countable dense set \(S\subseteq X\) such that for all \(i\in\omega\) and for all but finitely many \(s\in S\), \(s\in D_i\).
(\(DP\) is form [8 N]).
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\(DP^*\) ([126 E]): If \((X,T)\) is a topological space, then for every family
\(D=\{D_i: i\in\omega\}\) of dense open sets of \(X\), there is a countable dense set \(S\subseteq X\) such that for all \(i\in\omega\)
and for all but finitely many \(s\in S\), \(s\in D_i\).
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\(WCMC\) ([126 D]): For every denumerable family \(A\) of disjoint non-empty sets there is a
set \(C\subset\bigcup A\) such that for every \(a\in A\) ,\(0\le|C\cap a|<\omega\).
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\(FSCB\): Every separable first countable space \((X,T)\) has a countable \(\pi\)-base.
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\(Matrix(\omega)\): For every atomless poset \((P,\le)\) and every family \(D=\{D_i: i\in\omega\}\) of dense open subsets of \(P\), there exists a
matrix \(C=\{C_i: i\in\omega\}\), \(|\bigcup C|=\omega\) of \(P\) for \(D\). (\(Matrix(\omega)\) is form
[43 Q].)
Results:
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\(MC(\aleph_0,\infty)\) (Form 126) \(\leftrightarrow MC_{\omega}(\aleph_0,\infty) + MC(\aleph_0,\aleph_0)\)
(Form 360). (The conjunction \(MC_{\omega}(\aleph_0,\infty) + MC(\aleph_0,\aleph_0)\) is form
[126 B].)
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\(C(\aleph_0,\infty)\) (Form 8) \(\leftrightarrow MC_{\omega}(\aleph_0,\infty) + C(\aleph_0,\aleph_0)\)
(Form 32).(The conjunction \(MC_{\omega}(\aleph_0,\infty) + C(\aleph_0,\aleph_0)\)is
form [8 M].)
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\(C(\aleph_0,\infty)\) (Form 8) \(\leftrightarrow DP\) (form
[8 N]).
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\(MC(\aleph_0,\infty)\) (Form 126)\(\leftrightarrow WCMC\)
([126 D]).
(Clearly, \(MC(\aleph_0,\infty)\to WCMC\). Let \(X=\{A_n: n\in\omega\}\) be a denumerable set of pairwise disjoint non-empty sets. For each
\(n\in\omega\), let \(B_n=A_0\times A_1\times\cdots\times A_n\) and let \(B=\{B_n: n\in\omega\}\). Let \(C=\{B_{n_i}: n_i\in\omega\}\) be a
denumerable subset of \(B\) which has a multiple choice function \(f\). The function \(f\) can be used to construct a choice function on \(A\).
This result was noted by K. Keremedis.)
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\(WCMC\) ([126 D])\(\leftrightarrow DP^*\)
([126 E])\(\leftrightarrow pseudo(\omega)\)
([126 F]).
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\(MC(\aleph_0,\infty)\) (Form 126) \(\leftrightarrow FSCB + MC_{\omega}(\aleph_0,\infty)\).
(The conjunction \(FSCB +MC_{\omega}(\aleph_0,\infty)\) is form [126 C].)
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\(DC(\omega)\) (the Principle of Dependent Choices (Form 43)) \(\leftrightarrow Matrix(\omega)\)
(form [43 Q]).
Howard-Rubin number:
132
Type:
Definitions and results
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