In this paper we discuss how to define an appropriate notion of weak topology in the Wasserstein space $(\mathcal{P}_2(\mathsf{H}),W_2)$ Borel probability measures with finite quadratic moment on a separable Hilbert $\mathsf{H}$.

We state and prove that a certain class of smooth functions said to be BH-separable is meagre subset for the Fr\'echet topology. Because these are only admissible Hamiltonians Arnold-Liouville systems admitting bi-Hamiltonian structure, we get that, generically, cannot bi-Hamiltonian. At end paper, determine, both as concrete representation our general result an illustrative list, which polynom...

Kripke’s schema with parameters turns out to be equivalent to each of the following two statements from metric topology: every open subspace of a separable metric space is separable; every open subset of a separable metric space is a countable union of open balls. Thus Kripke’s schema serves as a point of reference for classifying theorems of classical mathematics within Bishop-style constructi...

We define a new topology, weaker than the gap topology, on the space of selfadjoint unbounded operators on a separable Hilbert space. We show that the subspace of selfadjoint Fredholm operators represents the functor K from the category of compact spaces to the category of abelian groups and prove a similar result for K. We define the spectral flow of a continuous path of selfadjoint Fredholm o...

This paper is an investigation of the universal separable metric space up to isometry U discovered by Urysohn. A concrete construction of U as a metric subspace of the space C[0, 1] of functions from [0, 1] to the reals with the supremum metric is given. An answer is given to a question of Sierpiński on isometric embeddings of U in C[0, 1]. It is shown that the closed linear span of an isometri...

If a sequence of random closed sets Xn in a separable complete metric space converges in distribution in the Wijsman topology to X, then the corresponding sequence of cores (sets of probability measures dominated by the capacity functional of Xn) converges to the core of the capacity of X. Core convergence is achieved not only in the Wijsman topology, but even in the stronger Vietoris topology....

In the present work, we give necessary and sufficient conditions on the graph of a right-angled Artin group that determine whether the group is subgroup separable or not. Also we show that right-angled Artin groups are residually torsion-free nilpotent. Moreover, we investigate the profinite topology of F2 × F2 and of the group L in [18], which are the only obstructions for the subgroup separab...

Proof. Let 11 and V be countable bases for X and Y, respectively, and let 3 have as sub-base the collection of sets {fCC(X, Y) \f(U) C V), with UCM and VCÜ. Then 3 certainly has a countable base. Clearly 3 also makes (/, x)—>/(x) jointly continuous, and hence [l, p. 223] is finer than the compact-open topology. Finally, every subset of C(X, Y) is certainly Lindelöf and separable for the countab...

In this article we give a purely topological characterization for a topology 3 on a set X to be the order topology with respect to some linear order R on X, as follows. A topology 3 on a set X is an order topology iff (X, 3) is a TVspace and 3 is the least upper bound of two minimal 7Vtopologies [Theorem 1 ]. From this we deduce a purely topological description of the usual topology on the set ...