Amorphe Potenzen kompakter Räume
Abstract A set is amorphous, if it is not a union of two disjoint infinite subsets. The following variants of the Tychonoff product theorem are investigated in the hierarchy of weak choice principles. TA1: An amorphous power of a compactT 2 space is compact. TA2: An amorphous power of a compactT 2 s...
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E-Artikel |
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| Sprache: |
Deutsch |
| Erschienen: |
1984 |
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| Umfang: |
17 |
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Springer Online Journal Archives 1860-2002 |
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| Übergeordnetes Werk: |
in: Archive for mathematical logic - 1950, 24(1984) vom: Jan., Seite 119-135 |
| Übergeordnetes Werk: |
volume:24 ; year:1984 ; month:01 ; pages:119-135 ; extent:17 |
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| 520 | |a Abstract A set is amorphous, if it is not a union of two disjoint infinite subsets. The following variants of the Tychonoff product theorem are investigated in the hierarchy of weak choice principles. TA1: An amorphous power of a compactT 2 space is compact. TA2: An amorphous power of a compactT 2 space which as a set is wellorderable is compact. In ZF0TA1 is equivalent to the assertion, that amorphous sets are finite. RT is Ramsey's theorem, that every finite colouring of the set ofn-element subsets of an infinite set has an infinite homogeneous subset and PW is Rubin's axiom, that the power set of an ordinal is wellorderable. In ZF0RT+PW implies TA2. Since RT+PW is compatible with the existence of infinite amorphous sets, TA2 does not imply TA1 in ZF0. But TA2 cannot be proved in ZF0 alone. As an application, we prove a theorem of Stone, using a weak wellordering axiomD 3 (a set is wellorderable, if each of its infinite subsets is structured) together with RT. | ||
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(DE-627)NLEJ200053426 DE-627 ger DE-627 rakwb ger Amorphe Potenzen kompakter Räume 1984 17 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier Abstract A set is amorphous, if it is not a union of two disjoint infinite subsets. The following variants of the Tychonoff product theorem are investigated in the hierarchy of weak choice principles. TA1: An amorphous power of a compactT 2 space is compact. TA2: An amorphous power of a compactT 2 space which as a set is wellorderable is compact. In ZF0TA1 is equivalent to the assertion, that amorphous sets are finite. RT is Ramsey's theorem, that every finite colouring of the set ofn-element subsets of an infinite set has an infinite homogeneous subset and PW is Rubin's axiom, that the power set of an ordinal is wellorderable. In ZF0RT+PW implies TA2. Since RT+PW is compatible with the existence of infinite amorphous sets, TA2 does not imply TA1 in ZF0. But TA2 cannot be proved in ZF0 alone. As an application, we prove a theorem of Stone, using a weak wellordering axiomD 3 (a set is wellorderable, if each of its infinite subsets is structured) together with RT. Springer Online Journal Archives 1860-2002 Brunner, Norbert oth in Archive for mathematical logic 1950 24(1984) vom: Jan., Seite 119-135 (DE-627)NLEJ188992863 (DE-600)1398309-x 1432-0665 nnns volume:24 year:1984 month:01 pages:119-135 extent:17 http://dx.doi.org/10.1007/BF02007144 GBV_USEFLAG_U ZDB-1-SOJ GBV_NL_ARTICLE AR 24 1984 1 119-135 17 |
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(DE-627)NLEJ200053426 DE-627 ger DE-627 rakwb ger Amorphe Potenzen kompakter Räume 1984 17 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier Abstract A set is amorphous, if it is not a union of two disjoint infinite subsets. The following variants of the Tychonoff product theorem are investigated in the hierarchy of weak choice principles. TA1: An amorphous power of a compactT 2 space is compact. TA2: An amorphous power of a compactT 2 space which as a set is wellorderable is compact. In ZF0TA1 is equivalent to the assertion, that amorphous sets are finite. RT is Ramsey's theorem, that every finite colouring of the set ofn-element subsets of an infinite set has an infinite homogeneous subset and PW is Rubin's axiom, that the power set of an ordinal is wellorderable. In ZF0RT+PW implies TA2. Since RT+PW is compatible with the existence of infinite amorphous sets, TA2 does not imply TA1 in ZF0. But TA2 cannot be proved in ZF0 alone. As an application, we prove a theorem of Stone, using a weak wellordering axiomD 3 (a set is wellorderable, if each of its infinite subsets is structured) together with RT. Springer Online Journal Archives 1860-2002 Brunner, Norbert oth in Archive for mathematical logic 1950 24(1984) vom: Jan., Seite 119-135 (DE-627)NLEJ188992863 (DE-600)1398309-x 1432-0665 nnns volume:24 year:1984 month:01 pages:119-135 extent:17 http://dx.doi.org/10.1007/BF02007144 GBV_USEFLAG_U ZDB-1-SOJ GBV_NL_ARTICLE AR 24 1984 1 119-135 17 |
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(DE-627)NLEJ200053426 DE-627 ger DE-627 rakwb ger Amorphe Potenzen kompakter Räume 1984 17 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier Abstract A set is amorphous, if it is not a union of two disjoint infinite subsets. The following variants of the Tychonoff product theorem are investigated in the hierarchy of weak choice principles. TA1: An amorphous power of a compactT 2 space is compact. TA2: An amorphous power of a compactT 2 space which as a set is wellorderable is compact. In ZF0TA1 is equivalent to the assertion, that amorphous sets are finite. RT is Ramsey's theorem, that every finite colouring of the set ofn-element subsets of an infinite set has an infinite homogeneous subset and PW is Rubin's axiom, that the power set of an ordinal is wellorderable. In ZF0RT+PW implies TA2. Since RT+PW is compatible with the existence of infinite amorphous sets, TA2 does not imply TA1 in ZF0. But TA2 cannot be proved in ZF0 alone. As an application, we prove a theorem of Stone, using a weak wellordering axiomD 3 (a set is wellorderable, if each of its infinite subsets is structured) together with RT. Springer Online Journal Archives 1860-2002 Brunner, Norbert oth in Archive for mathematical logic 1950 24(1984) vom: Jan., Seite 119-135 (DE-627)NLEJ188992863 (DE-600)1398309-x 1432-0665 nnns volume:24 year:1984 month:01 pages:119-135 extent:17 http://dx.doi.org/10.1007/BF02007144 GBV_USEFLAG_U ZDB-1-SOJ GBV_NL_ARTICLE AR 24 1984 1 119-135 17 |
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(DE-627)NLEJ200053426 DE-627 ger DE-627 rakwb ger Amorphe Potenzen kompakter Räume 1984 17 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier Abstract A set is amorphous, if it is not a union of two disjoint infinite subsets. The following variants of the Tychonoff product theorem are investigated in the hierarchy of weak choice principles. TA1: An amorphous power of a compactT 2 space is compact. TA2: An amorphous power of a compactT 2 space which as a set is wellorderable is compact. In ZF0TA1 is equivalent to the assertion, that amorphous sets are finite. RT is Ramsey's theorem, that every finite colouring of the set ofn-element subsets of an infinite set has an infinite homogeneous subset and PW is Rubin's axiom, that the power set of an ordinal is wellorderable. In ZF0RT+PW implies TA2. Since RT+PW is compatible with the existence of infinite amorphous sets, TA2 does not imply TA1 in ZF0. But TA2 cannot be proved in ZF0 alone. As an application, we prove a theorem of Stone, using a weak wellordering axiomD 3 (a set is wellorderable, if each of its infinite subsets is structured) together with RT. Springer Online Journal Archives 1860-2002 Brunner, Norbert oth in Archive for mathematical logic 1950 24(1984) vom: Jan., Seite 119-135 (DE-627)NLEJ188992863 (DE-600)1398309-x 1432-0665 nnns volume:24 year:1984 month:01 pages:119-135 extent:17 http://dx.doi.org/10.1007/BF02007144 GBV_USEFLAG_U ZDB-1-SOJ GBV_NL_ARTICLE AR 24 1984 1 119-135 17 |
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(DE-627)NLEJ200053426 DE-627 ger DE-627 rakwb ger Amorphe Potenzen kompakter Räume 1984 17 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier Abstract A set is amorphous, if it is not a union of two disjoint infinite subsets. The following variants of the Tychonoff product theorem are investigated in the hierarchy of weak choice principles. TA1: An amorphous power of a compactT 2 space is compact. TA2: An amorphous power of a compactT 2 space which as a set is wellorderable is compact. In ZF0TA1 is equivalent to the assertion, that amorphous sets are finite. RT is Ramsey's theorem, that every finite colouring of the set ofn-element subsets of an infinite set has an infinite homogeneous subset and PW is Rubin's axiom, that the power set of an ordinal is wellorderable. In ZF0RT+PW implies TA2. Since RT+PW is compatible with the existence of infinite amorphous sets, TA2 does not imply TA1 in ZF0. But TA2 cannot be proved in ZF0 alone. As an application, we prove a theorem of Stone, using a weak wellordering axiomD 3 (a set is wellorderable, if each of its infinite subsets is structured) together with RT. Springer Online Journal Archives 1860-2002 Brunner, Norbert oth in Archive for mathematical logic 1950 24(1984) vom: Jan., Seite 119-135 (DE-627)NLEJ188992863 (DE-600)1398309-x 1432-0665 nnns volume:24 year:1984 month:01 pages:119-135 extent:17 http://dx.doi.org/10.1007/BF02007144 GBV_USEFLAG_U ZDB-1-SOJ GBV_NL_ARTICLE AR 24 1984 1 119-135 17 |
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Abstract A set is amorphous, if it is not a union of two disjoint infinite subsets. The following variants of the Tychonoff product theorem are investigated in the hierarchy of weak choice principles. TA1: An amorphous power of a compactT 2 space is compact. TA2: An amorphous power of a compactT 2 space which as a set is wellorderable is compact. In ZF0TA1 is equivalent to the assertion, that amorphous sets are finite. RT is Ramsey's theorem, that every finite colouring of the set ofn-element subsets of an infinite set has an infinite homogeneous subset and PW is Rubin's axiom, that the power set of an ordinal is wellorderable. In ZF0RT+PW implies TA2. Since RT+PW is compatible with the existence of infinite amorphous sets, TA2 does not imply TA1 in ZF0. But TA2 cannot be proved in ZF0 alone. As an application, we prove a theorem of Stone, using a weak wellordering axiomD 3 (a set is wellorderable, if each of its infinite subsets is structured) together with RT. |
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Abstract A set is amorphous, if it is not a union of two disjoint infinite subsets. The following variants of the Tychonoff product theorem are investigated in the hierarchy of weak choice principles. TA1: An amorphous power of a compactT 2 space is compact. TA2: An amorphous power of a compactT 2 space which as a set is wellorderable is compact. In ZF0TA1 is equivalent to the assertion, that amorphous sets are finite. RT is Ramsey's theorem, that every finite colouring of the set ofn-element subsets of an infinite set has an infinite homogeneous subset and PW is Rubin's axiom, that the power set of an ordinal is wellorderable. In ZF0RT+PW implies TA2. Since RT+PW is compatible with the existence of infinite amorphous sets, TA2 does not imply TA1 in ZF0. But TA2 cannot be proved in ZF0 alone. As an application, we prove a theorem of Stone, using a weak wellordering axiomD 3 (a set is wellorderable, if each of its infinite subsets is structured) together with RT. |
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Abstract A set is amorphous, if it is not a union of two disjoint infinite subsets. The following variants of the Tychonoff product theorem are investigated in the hierarchy of weak choice principles. TA1: An amorphous power of a compactT 2 space is compact. TA2: An amorphous power of a compactT 2 space which as a set is wellorderable is compact. In ZF0TA1 is equivalent to the assertion, that amorphous sets are finite. RT is Ramsey's theorem, that every finite colouring of the set ofn-element subsets of an infinite set has an infinite homogeneous subset and PW is Rubin's axiom, that the power set of an ordinal is wellorderable. In ZF0RT+PW implies TA2. Since RT+PW is compatible with the existence of infinite amorphous sets, TA2 does not imply TA1 in ZF0. But TA2 cannot be proved in ZF0 alone. As an application, we prove a theorem of Stone, using a weak wellordering axiomD 3 (a set is wellorderable, if each of its infinite subsets is structured) together with RT. |
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