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A completion of ℤ is a field

José E. Marcos (2003)

Czechoslovak Mathematical Journal

We define various ring sequential convergences on ℤ and ℚ . We describe their properties and properties of their convergence completions. In particular, we define a convergence 𝕃 1 on ℤ by means of a nonprincipal ultrafilter on the positive prime numbers such that the underlying set of the completion is the ultraproduct of the prime finite fields ℤ / ( p ) . Further, we show that ( ℤ , 𝕃 1 * ) is sequentially precompact but fails to be strongly sequentially precompact; this solves a problem posed by D. Dikranjan.

A continuous operator extending fuzzy ultrametrics

I. Stasyuk, Edward D. Tymchatyn (2011)

Commentationes Mathematicae Universitatis Carolinae

We consider the problem of simultaneous extension of fuzzy ultrametrics defined on closed subsets of a complete fuzzy ultrametric space. We construct an extension operator that preserves the operation of pointwise minimum of fuzzy ultrametrics with common domain and an operation which is an analogue of multiplication by a constant defined for fuzzy ultrametrics. We prove that the restriction of the extension operator onto the set of continuous, partial fuzzy ultrametrics is continuous with respect...

A convergence on Boolean algebras generalizing the convergence on the Aleksandrov cube

Miloš S. Kurilić, Aleksandar Pavlović (2014)

Czechoslovak Mathematical Journal

We compare the forcing-related properties of a complete Boolean algebra 𝔹 with the properties of the convergences λ s (the algebraic convergence) and λ ls on 𝔹 generalizing the convergence on the Cantor and Aleksandrov cube, respectively. In particular, we show that λ ls is a topological convergence iff forcing by 𝔹 does not produce new reals and that λ ls is weakly topological if 𝔹 satisfies condition ( ℏ ) (implied by the 𝔱 -cc). On the other hand, if λ ls is a weakly topological convergence, then 𝔹 is a 2 𝔥 -cc algebra...

A Corson compact L-space from a Suslin tree

Peter Nyikos (2015)

Colloquium Mathematicae

The completion of a Suslin tree is shown to be a consistent example of a Corson compact L-space when endowed with the coarse wedge topology. The example has the further properties of being zero-dimensional and monotonically normal.

A generalization of a generic theorem in the theory of cardinal invariants of topological spaces

Alejandro Ramírez-Páramo, Noé Trinidad Tapia-Bonilla (2007)

Commentationes Mathematicae Universitatis Carolinae

The main goal of this paper is to establish a technical result, which provides an algorithm to prove several cardinal inequalities and relative versions of cardinal inequalities related to the well-known Arhangel’skii’s inequality: If X is a T 2 -space, then | X | ≤ 2 L ( X ) χ ( X ) . Moreover, we will show relative versions of three well-known cardinal inequalities.

A generalization of Čech-complete spaces and Lindelöf Σ -spaces

Aleksander V. Arhangel'skii (2013)

Commentationes Mathematicae Universitatis Carolinae

The class of s -spaces is studied in detail. It includes, in particular, all Čech-complete spaces, Lindelöf p -spaces, metrizable spaces with the weight ≤ 2 ω , but countable non-metrizable spaces and some metrizable spaces are not in it. It is shown that s -spaces are in a duality with Lindelöf Σ -spaces: X is an s -space if and only if some (every) remainder of X in a compactification is a Lindelöf Σ -space [Arhangel’skii A.V., Remainders of metrizable and close to metrizable spaces, Fund. Math. 220 (2013),...

A generic theorem in the theory of cardinal invariants of topological spaces

Aleksander V. Arhangel'skii (1995)

Commentationes Mathematicae Universitatis Carolinae

Relative versions of many important theorems on cardinal invariants of topological spaces are formulated and proved on the basis of a general technical result, which provides an algorithm for such proofs. New relative cardinal invariants are defined, and open problems are discussed.

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