Metrizable spaces where the inductive dimensions disagree. (English) Zbl 0724.54034
The author constructed the following examples. Note that a metric space X is called a gap metric space when it satisfies ind X\(<Ind X.\)
(1) There exists a gap metric space with its weight w(X) of cardinality continuum \({\mathfrak c}\), which is complete metrizable and N-compact. (2) There exists a gap metric space with \(w(X)={\mathfrak c}\), which is a subspace of Roy’s gap space \(\Delta\), and N-compact. (3) There exists a gap metric space with \(w(X)={\mathfrak c}\), which contains no gap metric subspaces of weight less than \({\mathfrak c}\). - Example (1) answers a question of Mrowka in [Rings of continuous functions (1985; Zbl 0559.00006), p. 313, problem 6] affirmatively. (2) answers a question of Roy, “Is every gap subspace of \(\Delta\) non-N-compact?”, negatively. Example (3) answers the question “Does every gap metric space contain a gap metric space of weight \(\omega_ 1?''\) negatively, assuming the negation of CH. Related with this problem a version of (3) is constructed for every infinite cardinal \(\lambda\) under the following additional axiom: For a cardinal \(\kappa\), E(\(\kappa\)) is the statement: There is a set E of ordinals of cofinality \(\omega\) which is stationary in \(\kappa\), but for all \(\beta <\kappa\), \(E\cap \beta\) is not stationary in \(\beta\). The following statement \(E^*\) is known to be consistent with the usual axiom of set theory. \(E^*:\) For all singular strong limit cardinals \(\kappa,2^{\kappa}=\kappa^+\) and \(E(\kappa^+).\)
Theorem (Assume \(E^*)\). For each cardinal \(\lambda\) there is a metrizable space \(X(\lambda)\) with \(Ind(X(\lambda))=1\) and for all \(Y\subset X(\lambda)\), with \(w(Y)\leq \lambda\), it follows that \(Ind(Y)=0.\)
Since the weight of the space \(X(\lambda)\) is \(\kappa^+\), where \(\kappa\) is a strong limit cardinal greater than \(\lambda\), there remains a problem of another version of (2). Problem. For each cardinal \(\lambda\) is there a metrizable space \(X(\lambda)\) with \(Ind(X(\lambda))=1\) and \(w(X(\lambda))=\lambda\), which satisfies that for all \(Y\subset X(\lambda)\), with \(w(Y)<\lambda\), it holds that \(Ind(Y)=0?\) Of course, there still remains the following famous problem [Rings of continuous functions (loc. cit.), p. 312, The main problem 1] after Roy’s paper: Does there exist a gap metric space X with \(\dim X-ind X>1\)?
(1) There exists a gap metric space with its weight w(X) of cardinality continuum \({\mathfrak c}\), which is complete metrizable and N-compact. (2) There exists a gap metric space with \(w(X)={\mathfrak c}\), which is a subspace of Roy’s gap space \(\Delta\), and N-compact. (3) There exists a gap metric space with \(w(X)={\mathfrak c}\), which contains no gap metric subspaces of weight less than \({\mathfrak c}\). - Example (1) answers a question of Mrowka in [Rings of continuous functions (1985; Zbl 0559.00006), p. 313, problem 6] affirmatively. (2) answers a question of Roy, “Is every gap subspace of \(\Delta\) non-N-compact?”, negatively. Example (3) answers the question “Does every gap metric space contain a gap metric space of weight \(\omega_ 1?''\) negatively, assuming the negation of CH. Related with this problem a version of (3) is constructed for every infinite cardinal \(\lambda\) under the following additional axiom: For a cardinal \(\kappa\), E(\(\kappa\)) is the statement: There is a set E of ordinals of cofinality \(\omega\) which is stationary in \(\kappa\), but for all \(\beta <\kappa\), \(E\cap \beta\) is not stationary in \(\beta\). The following statement \(E^*\) is known to be consistent with the usual axiom of set theory. \(E^*:\) For all singular strong limit cardinals \(\kappa,2^{\kappa}=\kappa^+\) and \(E(\kappa^+).\)
Theorem (Assume \(E^*)\). For each cardinal \(\lambda\) there is a metrizable space \(X(\lambda)\) with \(Ind(X(\lambda))=1\) and for all \(Y\subset X(\lambda)\), with \(w(Y)\leq \lambda\), it follows that \(Ind(Y)=0.\)
Since the weight of the space \(X(\lambda)\) is \(\kappa^+\), where \(\kappa\) is a strong limit cardinal greater than \(\lambda\), there remains a problem of another version of (2). Problem. For each cardinal \(\lambda\) is there a metrizable space \(X(\lambda)\) with \(Ind(X(\lambda))=1\) and \(w(X(\lambda))=\lambda\), which satisfies that for all \(Y\subset X(\lambda)\), with \(w(Y)<\lambda\), it holds that \(Ind(Y)=0?\) Of course, there still remains the following famous problem [Rings of continuous functions (loc. cit.), p. 312, The main problem 1] after Roy’s paper: Does there exist a gap metric space X with \(\dim X-ind X>1\)?
Reviewer: K.Tsuda (Bunkyo-chou, Matsuyama)
Citations:
Zbl 0559.00006References:
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