Form Classes

Statement:

\(MC_\omega(\infty,\infty)\) (\(\omega\)-MC): For every family \(X\) of pairwise disjoint non-empty sets, there is a function \(f\) such that for each \(x\in X\), f(x) is a non-empty countable subset of \(x\).

Howard_Rubin_Number: 76

Parameter(s): This form does not depend on parameters

This form's transferability is: Unknown

This form's negation transferability is: Negation Transferable

Article Citations:
Howard-Rubin-Keremedis-Rubin-1998b: Disjoint unions of topological spaces and choice

Book references

Note connections:

The following forms are listed as conclusions of this form class in rfb1: 173, 425, 131,

Back

Complete List of Equivalent Forms

Howard-Rubin Number Statement References
76 A

Every open cover \(\cal{U}\) of a metric space can be written as a well ordered union \(\bigcup\{U_\alpha:\alpha\in\gamma\}\) where \(\gamma\) is an ordinal and each \(U_\alpha\) is locally countable.

Howard-Rubin-Stanley-Keremedis-2000a
Note [141]
76 B

Every open cover \(\cal{U}\) of a metric space can be written as a well ordered union \(\bigcup\{U_\alpha:\alpha\in\gamma\}\) where \(\gamma\) is an ordinal and each \(U_\alpha\) is point countable.

Howard-Rubin-Stanley-Keremedis-2000a
Note [141]