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In pointfree topology, the notion of uniformity in the form of a system of covers was introduced by J. Isbell in [11], and later developed by A. Pultr in [14] and [15]. Another equivalent notion of locale uniformity was given by P. Fletcher and W. Hunsaker in [6], which they called “entourage uniformity”. The purpose of this paper is to formulate and investigate an alternative definition of entourage uniformity which is more likely to the Weil pointed entourage uniformity, since it is expressed...
The following two theorems give the flavour of what will be proved. Theorem. Let Y be a complete metric space. Then the families of first Baire class functions and of first Borel class functions from [0,1] to Y coincide if and only if Y is connected and locally connected.Theorem. Let Y be a separable metric space. Then the families of second Baire class functions and of second Borel class functions from [0,1] to Y coincide if and only if for all finite sequences of nonempty open subsets of Y there...
Let M be a metrizable group. Let G be a dense subgroup of . We prove that if G is domain representable, then . The following corollaries answer open questions. If X is completely regular and is domain representable, then X is discrete. If X is zero-dimensional, T₂, and is subcompact, then X is discrete.
We study transitive non-minimal ℕ-actions and ℤ-actions. We show that there are such actions whose non-transitive points are periodic and whose topological entropy is positive. It turns out that such actions can be obtained by perturbing minimal systems under some reasonable assumptions.
Theorem. In ZF (i.e., Zermelo-Fraenkel set theory without the axiom of choice) the following conditions are equivalent: (1) is a Lindelöf space, (2) is a Lindelöf space, (3) is a Lindelöf space, (4) every topological space with a countable base is a Lindelöf space, (5) every subspace of is separable, (6) in , a point is in the closure of a set iff there exists a sequence in that converges to , (7) a function is continuous at a point iff is sequentially continuous at , (8)...
Let be a completely regular Hausdorff space and, as usual, let denote the ring of real-valued continuous functions on . The lattice of -ideals of has been shown by Martínez and Zenk (2005) to be a frame. We show that the spectrum of this lattice is (homeomorphic to) precisely when is a -space. This we actually show to be true not only in spaces, but in locales as well. Recall that an ideal of a commutative ring is called a -ideal if whenever two elements have the same annihilator and...
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