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On \emptyset-definable elements in a field
Apoloniusz Tyszka
Let K be a field and \tilde{K} denote the set of all r \in K for which there exists a finite set A(r) with {r} \subseteq A(r) \subseteq K such that each mapping f:A(r) \to K that s…
A stronger form of the theorem constructing a rigid binary relation on any set
Apoloniusz Tyszka
On every set A there is a rigid binary relation i.e. such a relation R \subseteq A \times A that there is no homomorphism (A,R) \rightarrow (A,R) except the identity (Vop{ě}nka et…
On binary relations without non-identical endomorphisms
Apoloniusz Tyszka
On every set A there is a rigid binary relation, i.e. such a relation R that there is no homomorphism (A,R)->(A,R) except the identity (Vopenka et al. [1965]). We state two conject…
A combinatorial characterization of second category subsets of X^ω
Apoloniusz Tyszka
Let a finite non-empty X is equipped with discrete topology. We prove that S \subseteq X^ωis of second category if and only if for each f:ω-> \bigcup_{n \in ω} X^n there exists a s…
A new combinatorial characterization of the minimal cardinality of a subset of R which is not of first category
Apoloniusz Tyszka
Let M denote the ideal of first category subsets of R. We prove that min{card X: X \subseteq R, X \not\in M} is the smallest cardinality of a family S \subseteq {0,1}^ωwith the pro…