On two open problems of contractive mappings.
On two recurrence problems
We review some aspects of recurrence in topological dynamics and focus on two open problems. The first is an old one concerning the relation between Poincaré and Birkhoff recurrence; the second, due to the first author, is about moving recurrence. We provide a partial answer to a topological version of the moving recurrence problem.
On ultrametric space.
On uniformly generalized Lipschitzian mappings.
On universality of countable and weak products of sigma hereditarily disconnected spaces
Suppose a metrizable separable space Y is sigma hereditarily disconnected, i.e., it is a countable union of hereditarily disconnected subspaces. We prove that the countable power of any subspace X ⊂ Y is not universal for the class ₂ of absolute -sets; moreover, if Y is an absolute -set, then contains no closed topological copy of the Nagata space = W(I,ℙ); if Y is an absolute -set, then contains no closed copy of the Smirnov space σ = W(I,0). On the other hand, the countable power of...
On universality of finite powers of locally path-connected meager spaces
It is shown that for every integer n the (2n+1)th power of any locally path-connected metrizable space of the first Baire category is 𝓐₁[n]-universal, i.e., contains a closed topological copy of each at most n-dimensional metrizable σ-compact space. Also a one-dimensional σ-compact absolute retract X is found such that the power X^{n+1} is 𝓐₁[n]-universal for every n.
On vector measures which have everywhere infinite variation or noncompact range [Book]
On weak convergence to the fixed point of a generalized asymptotically nonexpansive map.
On weakly* conditionally compact dynamical systems
On Weakly Measurable Functions
We show that if T is an uncountable Polish space, 𝓧 is a metrizable space and f:T→ 𝓧 is a weakly Baire measurable function, then we can find a meagre set M ⊆ T such that f[T∖M] is a separable space. We also give an example showing that "metrizable" cannot be replaced by "normal".
On -embedded sets and extension of mappings
We introduce and study -embedded sets and apply them to generalize the Kuratowski Extension Theorem.
On -generalized closed sets.
On -discrete Borel mappings via quasi-metrics
On -porous sets in abstract spaces.
On -fixed point for maps on uniform spaces.
One tripled fixed point result in partially ordered 0-complete partial metric spaces
One-dimensional chain recurrent sets of flows in the 2-sphere.
One-Parameter Flows with the Pseudo Orbit Tracing Property.
One-to one Carathéodory representation theorem for multifunctions with uncountable values
Only 3-generalized metric spaces have a compatible symmetric topology
We prove that every 3-generalized metric space is metrizable. We also show that for any ʋ with ʋ ≥ 4, not every ʋ-generalized metric space has a compatible symmetric topology.