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Normal numbers and subsets of N with given densities

Haseo Ki, Tom Linton (1994)

Fundamenta Mathematicae

For X ⊆ [0,1], let D X denote the collection of subsets of ℕ whose densities lie in X. Given the exact location of X in the Borel or difference hierarchy, we exhibit the exact location of D X . For α ≥ 3, X is properly D ξ ( Π α 0 ) iff D X is properly D ξ ( Π 1 + α 0 ) . We also show that for every nonempty set X ⊆[0,1], D X is Π 3 0 -hard. For each nonempty Π 2 0 set X ⊆ [0,1], in particular for X = x, D X is Π 3 0 -complete. For each n ≥ 2, the collection of real numbers that are normal or simply normal to base n is Π 3 0 -complete. Moreover, D ℚ , the...

Normal numbers and the Borel hierarchy

Verónica Becher, Pablo Ariel Heiber, Theodore A. Slaman (2014)

Fundamenta Mathematicae

We show that the set of absolutely normal numbers is Π⁰₃-complete in the Borel hierarchy of subsets of real numbers. Similarly, the set of absolutely normal numbers is Π⁰₃-complete in the effective Borel hierarchy.

Note on "construction of uninorms on bounded lattices"

Xiu-Juan Hua, Hua-Peng Zhang, Yao Ouyang (2021)

Kybernetika

In this note, we point out that Theorem 3.1 as well as Theorem 3.5 in G. D. Çaylı and F. Karaçal (Kybernetika 53 (2017), 394-417) contains a superfluous condition. We have also generalized them by using closure (interior, resp.) operators.

Notes on locally internal uninorm on bounded lattices

Gül Deniz Çaylı, Ümit Ertuğrul, Tuncay Köroğlu, Funda Karaçal (2017)

Kybernetika

In the study, we introduce the definition of a locally internal uninorm on an arbitrary bounded lattice L . We examine some properties of an idempotent and locally internal uninorm on an arbitrary bounded latice L , and investigate relationship between these operators. Moreover, some illustrative examples are added to show the connection between idempotent and locally internal uninorm.

Nowhere dense subsets and Booth's Lemma

Viacheslav I. Malykhin (1996)

Commentationes Mathematicae Universitatis Carolinae

The following statement is proved to be independent from [ LB + ¬ CH ] : ( * ) Let X be a Tychonoff space with c ( X ) ≤ ℵ 0 and π w ( X ) < ℭ . Then a union of less than ℭ of nowhere dense subsets of X is a union of not greater than π w ( X ) of nowhere dense subsets.

OCA and towers in 𝒫 ( ℕ ) / f i n

Ilijas Farah (1996)

Commentationes Mathematicae Universitatis Carolinae

We shall show that Open Coloring Axiom has different influence on the algebra 𝒫 ( ℕ ) / f i n than on ℕ ℕ / f i n . The tool used to accomplish this is forcing with a Suslin tree.

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