Axiom Of Choice

Statement:

If \((X,\cal T) \) is a first countable topological space and \((\cal B(x))_{x\in X}\) is a family such that for all \(x \in X\), \(\cal B(x)\) is a local base at \(x\), then there is a family \(( \cal V(x))_{x\in X}\) such that for every \(x \in X\), \(\cal V(x)\) is a countable local base at \(x\) and \(\cal V(x) \subseteq \cal B(x)\).

Howard_Rubin_Number: 426

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:
Gutierres-2004: On first and second countable spaces and the axiom of choice

Book references

Note connections:
Note 159

Definitions for forms [0 AV], [8 AP] through [8 AS], [94 X], and Form 424 from Gutierres [2004].

The following forms are listed as conclusions of this form class in rfb1: 76, 8, 1,

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Complete List of Equivalent Forms

Howard-Rubin Number	Statement	References
426 A	The conjunction of Form 76 (\(MC_\omega(\infty,\infty)\)) and Form 8 (\(C(\aleph_0,\infty)\)).	Gutierres [2004]
426 B	The conjunction of Form 76 (\(MC_\omega(\infty, \infty)\)) and Form 31 (\(UT(\aleph_0,\aleph_0,\aleph_0)\)).	Gutierres [2004]
426 C	The conjunction of Form 76 (\(MC_\omega(\infty,\infty)\)) and Form 32 (\(C(\aleph_0,\aleph_0)\)).	Gutierres [2004]

Form Classes

Complete List of Equivalent Forms