An Effective Construction of a Free z‐ultrafilter | Zendy

HERRLICH HORST | Zendy

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An Effective Construction of a Free z‐ultrafilter

Author(s) -

HERRLICH HORST

Publication year - 1996

Publication title -

annals of the new york academy of sciences

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 1.712

H-Index - 248

eISSN - 1749-6632

pISSN - 0077-8923

DOI - 10.1111/j.1749-6632.1996.tb49169.x

Subject(s) - ultrafilter , equivalence (formal languages) , mathematics , zero (linguistics) , compact space , discrete mathematics , set (abstract data type) , space (punctuation) , pure mathematics , combinatorics , topology (electrical circuits) , computer science , philosophy , linguistics , operating system , programming language

It is known that free ultrafilters cannot be constructed effectively (i.e., in ZF set theory without any additional choice principle). In this paper a nonconvergent, thus free, z ‐ultrafilter is constructed effectively in a suitable zero‐dimensional topological space X . Thus the implication C ‐compact ? B ‐compact (which under AC is an equivalence for completely regular spaces) fails to be an equivalence in ZF even for zero‐dimensional spaces, where B ‐compact (respectively, C ‐compact) means that every ultrafilter (respectively, z ‐ultrafilter) converges.

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