On local uniformities

Two concepts simplifying the inner structure of uniform spaces are examined, namely Ginsburg-Isbell concept of local fineness and that of functional local fineness due to Z. Frolík (for definitions see below). It is shown that these two concepts are equivalent. Some information on Ginsburg-Isbell derivatives of metric spaces and on the construction of locally fine coreflection is obtained.     

模糊拟邻近结构与模糊双拓扑

本文给出了模糊拟邻近结构的概念，并着重讨论了模糊拟邻近结构与模糊双拓扑，模糊拟邻近结构与模糊拟一致空间之间的相互联系。     

Quasi—uniformities on Lattice Implication Algebras

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Lattice uniformities and modular functions on orthomodular lattices

It is shown that the lattice of all exhaustive lattice uniformities on an orthomodular latticeL is isomorphic to the centre of a natural completion (of a quotient) ofL, and is thus a complete Boolean algebra. This is applied to prove a decomposition theorem for exhaustive modular functions on orthomodular lattices, which generalizes Traynor's decomposition theorem [14].     

Boolean algebras of lattice uniformities and decompositions of modular functions

We study decomposition theorems for modular functions on lattices and the relationship between such decompositions and lattice properties of a suitable system of uniformities. We give a purely topological characterization for the validity of a decomposition theorem of a certain type and examine when this topological condition is satisfied, namely when a system of lattice uniformities is a Boolean algebra consisting of permutable uniformities.      

Sphere Packing, III. Extremal Cases

This paper is the third in a series of six papers devoted to the proof of the Kepler conjecture, which asserts that no packing of congruent balls in three dimensions has density greater than the face-centered cubic packing. In the previous paper in this series, a continuous function f on a compact space was defined, certain points in the domain were conjectured to give the global maxima, and the relation between this conjecture and the Kepler conjecture was established. This paper shows that those points are indeed local maxima. Various approximations to f are developed, that will be used in subsequent papers to bound the value of the function f. The function f can be expressed as a sum of terms, indexed by regions on a unit sphere. Detailed estimates of the terms corresponding to triangular and quadrilateral regions are developed.     

Convergences,pretopologies and proximities induced by Cauchy and Riesz screens

We answer a question of Császár [12]: under which conditions a given pretopological closure or proximity can be induced by a Cauchy structure? We give a characterization for these closures and proximities using properties of convergences and nasses [14] induced by Cauchy structures. We prove also that the set of Cauchy screens inducing a given reciprocal convergence structure is a non empty interval of the set of Cauchy screens equipped with the usual inclusion order.