Characterizations of ℕ-compactness and realcompactness via ultrafilters in the absence of the axiom of choice
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Alireza Olfati
und
Eliza Wajch
Abstract
This article concerns the Herrlich-Chew theorem stating that a Hausdorff zero-dimensional space is ℕ-compact if and only if every clopen ultrafilter with the countable intersection property in this space is fixed. It also concerns Hewitt’s theorem stating that a Tychonoff space is realcompact if and only if every z-ultrafilter with the countable intersection property in this space is fixed. The axiom of choice was involved in the original proofs of these theorems. The aim of this article is to show that the Herrlich-Chew theorem is valid in ZF, but it is an open problem if Hewitt’s theorem can be false in a model of ZF. It is proved that Hewitt’s theorem is true in every model of ZF in which the countable axiom of multiple choice is satisfied. A modification of Hewitt’s theorem is given and proved true in ZF. Several applications of the results obtained are shown.
(Communicated by L’ubica Holá)
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Artikel in diesem Heft
- Weak differences, weak BCK-algebras and applications to some partial orders on rings
- Weakly κ-compact topological spaces
- On sharp radius estimates for S*(β) and a product function
- Maximal subextension and stability in m-capacity of maximal subextension of m-subharmonic functions with given boundary values
- Asymptotic behavior of fractional super-linear differential equations
- New and improved oscillation criteria of third-order half-linear delay differential equations via canonical transform
- Global dynamics of the system of difference equations
- Results on oscillatory properties of third-order functional difference equations with semi-canonical operators
- A new approach to metrical fixed point theorems
- Generalized Baker’s result and stability of functional equations using fixed point results
- Characterizations of ℕ-compactness and realcompactness via ultrafilters in the absence of the axiom of choice
- K-theory of oriented flag manifolds
- On certain observations on split continuity and cauchy split continuity
- On the generalized eta- and theta-transformation formulas as the Hecke modular relation
- On some selective star Lindelöf-type properties
