Implications

We have the following indirect implication of form equivalence classes:

36 \(\Rightarrow\) 47-n
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
36 \(\Rightarrow\) 62 On Loeb and weakly Loeb Hausdorff spaces, Tachtsis, E. 2000, Math. Japon.
62 \(\Rightarrow\) 121 clear
121 \(\Rightarrow\) 33-n clear
33-n \(\Rightarrow\) 47-n clear

Here are the links and statements of the form equivalence classes referenced above:

Howard-Rubin Number Statement
36:

Compact T\(_2\) spaces are Loeb. (A space is Loeb if the set of non-empty closed sets has a choice function.)

62:

\(C(\infty,< \aleph_{0})\):  Every set of non-empty finite  sets  has  a choice function.

121:

\(C(LO,<\aleph_{0})\): Every linearly ordered set of non-empty finite sets has a choice function.

33-n:

If \(n\in\omega-\{0,1\}\), \(C(LO,n)\):  Every linearly ordered set of \(n\) element sets has  a choice function.

47-n:

If \(n\in\omega-\{0,1\}\), \(C(WO,n)\): Every well ordered collection of \(n\)-element sets has a choice function.

Comment:

Back