Implications

We have the following indirect implication of form equivalence classes:

36 \(\Rightarrow\) 46-K
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 \(\Rightarrow\) 61 clear
61 \(\Rightarrow\) 46-K 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.

61:

\((\forall n\in\omega, n\ge 2\))\((C(\infty,n))\): For each \(n\in\omega\), \(n\ge 2\), every set of \(n\) element  sets has a choice function.

46-K:

If \(K\) is a finite subset of \(\omega-\{0,1\}\), \(C(\infty,K)\): For every \(n\in K\), every set of \(n\)-element sets has a choice function.

Comment:

Back