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A Hall Criterion for Countable Families of Sets
Author(s) -
Holz Michael,
Podewski KlausPeter,
Steffens Karsten
Publication year - 1980
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-21.1.1
Subject(s) - countable set , injective function , mathematics , recursion (computer science) , set (abstract data type) , combinatorics , function (biology) , discrete mathematics , computer science , algorithm , evolutionary biology , biology , programming language
Let F = ( F ( i )| i ɛ I ) be a countable family. By recursion we define subsets I α ( B ) of I and prove that there is an ordinal λ such that F has an injective choice function if and only if |B| ⩾ |∪{ I γ ( B )|γ < λ) | for every finite set B .