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We introduce a weak Gurov–Reshetnyak class and discuss its connections to a weak Muckenhoupt A_\infty condition and a weak reverse Hölder inequality in the setting of metric measure spaces with a doubling measure. A John–Nirenberg type lemma is shown for the weak Gurov–Reshetnyak class which gives a specific decay estimate for the oscillation of a function. It implies that a function in the weak Gurov–Reshetnyak class satisfies the weak reverse Hölder inequality. This comes with an upper bound for the reverse Hölder exponent depending on the Gurov–Reshetnyak parameter which allows the study of the asymptotic behavior of the exponent. [ABSTRACT FROM AUTHOR]

We give a permutation model in which Stone's theorem (every metric space is paracompact) is false and the Boolean Prime Ideal Theorem (every ideal in a Boolean algebra extends to a prime ideal) is true. The erring metric space in our model attains only rational distances and is not metacompact. Transfer theorems give the comparable independence in the Zermelo-Fraenkel setting, answering a question of Good, Tree, and Watson. [ABSTRACT FROM AUTHOR]

We explain how dynamical systems with generating partitions are symbolically expansive, namely symbolic counterparts of the expansive ones. Similar ideas allow the notions of symbolic equicontinuity, symbolic distality, symbolic N-expansivity, and symbolic shadowing property. We analyze dynamical systems with these properties in the circle. Indeed, we show that every symbolically N-expansive circle homeomorphism has finitely many periodic points. Moreover, if there are no wandering points, then the situation will depend on the rotation number. In the rational case the homeomorphism is symbolically equicontinuous with the symbolic shadowing property and, in the irrational case, the homeomorphism is symbolically expansive, symbolically distal, but not symbolically equicontinuous. We will also introduce a symbolic entropy and study its properties. [ABSTRACT FROM AUTHOR]

The tensor product of two p-harmonic functions is in general not p-harmonic, but we show that it is a quasiminimizer. More generally, we show that the tensor product of two quasiminimizers is a quasiminimizer. Similar results are also obtained for quasisuperminimizers and for tensor sums. This is done in weighted Rn with p-admissible weights. It is also shown that the tensor product of two p-admissible measures is p-admissible. This last result is generalized to metric spaces. [ABSTRACT FROM AUTHOR]

We prove that the property of a set-valued mapping F : X ⇉ Y to be locally metrically regular (and consequently, the properties of the mapping to be linearly open or pseudo-Lipschitz) is separably reducible by rich families of separable subspaces of X×Y. In fact, we prove that, moreover, this extends to computation of the functor reg F that associates with F the rates of local metric regularity of F near points of its graph. [ABSTRACT FROM AUTHOR]

Given a separable metrisable space X, and a group G of home- omorphisms of X, we introduce a topological property of the action G X which is equivalent to the existence of a G-invariant compatible metric on X. This extends a result of Marjanovic obtained under the additional assumption that X is locally compact. [ABSTRACT FROM AUTHOR]

Let (X, d) be a complete metric space. We prove that there is a continuous, linear extension operator from the space of all partial, continuous, bounded metrics with closed, bounded domains in X endowed with the Haus-dorff metric topology to the space of all continuous, bounded, metrics on X with the topology of uniform convergence on compact sets. This is a variant of the result of Tymchatyn and Zarichnyi for continuous metrics defined on closed, variable domains in a compact metric space. We get a similar result for the case of continuous real-valued functions. [ABSTRACT FROM AUTHOR]