Axiom Of Choice

Description: A summary of the definitions and results from Brunner [1983d]

Content:

A summary of the definitions and results from Brunner [1983d].

Brunner considers 16 properties a topological space \((X,{\underline{X}})\) may have:

W1:	\(\underline{X}\) is w.o. (well ordered);
W2:	\(X\) is w.o.;
D1:	\(X\) is covered by a w.o. family of closed, discrete sets.
D2:	\(X\) is covered by a w.o. family of discrete sets;
B1:	\(X\) has a w.o. base
B2:	\(\exists f: X\times W \rightarrow {\underline{X}}\) such that \(W\) is w.o. and \(f'(\{x\}\times W)\) is a neighborhood base at \(x\) for each \(x \in X\).
B3:	\(\exists f: X\times W\rightarrow\underline{X}\) such that \(W\) is w.o. and \(\forall x\in X\), \(\{x\} = \bigcap f'(\{x\}\times W)\)
L1:	Each open cover of \(X\) has a w.o. subcover;
L2:	Each open cover of \(X\) has a w.o. refinement;
S:	\(X\) has a dense, w.o. subset;
C:	Each family \(O \subseteq {\underline{ X}}\) of pairwise disjoint sets is w.o.
A1:	If \(O \subseteq {\underline{X}}\) covers \(X\), \(\exists f: X \rightarrow O\) such that \()\forall x \in X\), \(x \in f(x)\);
A2:	If \(O \subseteq {\underline{ X}}\) covers \(X\), \(\exists f: X \rightarrow {\underline{ X}}\) s.t. \(\forall x \in X\), \(x \in f(x)\) and \(f'X\) refines \(O\);
H1:	\(X\) is hereditarily \(A1\);
H2:	\(X\) is hereditarily \(A2\);
F:	\(X\) is a continuous finite to one image of an \(A1\) space.

Definition: If \(P\) is one of the above then

\(A(P)\) is the statement: ``Every topological space \((X,{\underline X})\) is P.''
\(A(P,Q)\) is the statement: ``If a topological space satisfies \(P\) then it satisfies \(Q\).''

All of the \(A(P)\)'s occur in the list of forms, but not all the \(A(P,Q)\)'s occur.

The table below, taken from Brunner [1983d], summarizes the results. In the table, 0 means that \(A(P,Q)\) (\(P\) the row heading and \(Q\) the column heading) is provable in \(ZF^{0}\). \(N\) means \(A(P,Q)\) is not provable in \(ZF^{0}\). \(AC\) (Form 1), \(MC\) (Form 67) and \(PW\) (Form 91) mean respectively that \(A(P,Q)\) is equivalent to \(AC,\ MC\) or \(PW\) in \(ZF^{0}\). And \(IMC\) or \(IPW\) mean that \(A(P,Q)\) implies \(MC\) or \(PW\) respectively in \(ZF^{0}\).

Here's the table:
\[ \begin{array}{lc|ccccccccccccccccc} \\ & & & W1 & W2 & D1 & D2 & B1 & B2 & B3 & L1 & L2 & S & C & H1 & H2 & A1 & F & A2 \\ \hline & W1 && 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ & W2 && PW & 0 & 0 & 0 & ? & ? & 0 & N & ? & 0 & 0 & N & ? & N & N & ?\\ & D1 && AC & AC & 0 & 0 & AC & N & 0 & AC & AC & AC & AC & AC & N & AC & MC & N\\ &D2 && AC & AC &IMC& 0 & AC & MC & MC & AC & AC & AC & AC & AC & MC & AC & MC & MC \\ &B1 && PW &IPW&IPW&IPW& 0 & 0 & 0 & N & 0 & N & 0 & N & 0 & N & N & 0 \\ &B2 && AC & AC &IPW&IPW& AC & 0 & 0 & AC & AC & AC & AC & AC & 0 & AC & MC & 0 \\ &B3 && AC & AC &IPW&IPW& AC & N & 0 & AC & AC & AC & AC & AC & N & AC & MC & N \\ &L1 && AC & AC & MC & N & AC & MC & MC & 0 & 0 & AC & AC & AC & MC & 0 & 0 & 0 \\ &L2 && AC & AC & MC &IPW& AC & MC & MC & N & 0 & AC & AC & AC & MC & N & N & 0 \\ &S && PW &IPW&IPW&IPW& N & N & ? & N & N & 0 & 0 & N & N & N & N & N \\ &C &&IPW& PW & PW & PW & N & N & N & N & N & N & 0 & N & N & N & N & N \\ &H1 && N & N & N & ? & N & N & N & N & N & N & N & 0 & 0 & 0 & 0 & 0 \\ &H2 && AC & AC & AC &IPW& AC & MC & MC & AC & AC & AC & AC & AC & 0 & AC & MC & 0 \\ &A1 && AC & AC & MC & N & AC & MC & MC & N & N & AC & AC & AC & MC & 0 & 0 & 0 \\ &F && AC & AC & MC & N & AC & MC & MC & N & N & AC & AC & AC & MC & N & 0 & 0 \\ &A2 && AC & AC & MC &IPW& AC & MC & MC & AC & AC & AC & AC & AC & MC & AC & MC & 0 \\ \end{array} \]

Theorem

\(ZF \vdash A(P) \leftrightarrow AC\) for all \(P\) in the list.
\(ZF^0 \vdash A(P)\leftrightarrow AC\) for \(P \in \{W1,W2,B1,L1,L2,S,C,H1,A1\}.\)
\(ZF^0 \vdash A(P)\leftrightarrow MC\) ( Form 67) for \(P \in \{D1,B2,B3,H2,A2\}.\)
\(ZF^0 \vdash AC \rightarrow A(F) \rightarrow MC\) (Form 67) \( \rightarrow A(D2) \rightarrow PW\) ( Form 91).

Howard-Rubin number: 26

Type: Summary of definitions and results

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Notes