Ideal-quasi-Cauchy sequences

An ideal $I$ is a family of subsets of positive integers $\textbf{N}$ which is closed under taking finite unions and subsets of its elements. A sequence $(x_n)$ of real numbers is said to be $I$-convergent to a real number $L$, if for each \;$\varepsilon> 0$ the set $\{n:|x_{n}-L|\geq \varepsilon\}$ belongs to $I$. We introduce $I$-ward compactness of a subset of $\textbf{R}$, the set of real numbers, and $I$-ward continuity of a real function in the senses that a subset $E$ of $\textbf{R}$ is $I$-ward compact if any sequence $(x_{n})$ of points in $E$ has an $I$-quasi-Cauchy subsequence, and a real function is $I$-ward continuous if it preserves $I$-quasi-Cauchy sequences where a sequence $(x_{n})$ is called to be $I$-quasi-Cauchy when $(\Delta x_{n})$ is $I$-convergent to 0. We obtain results related to $I$-ward continuity, $I$-ward compactness, ward continuity, ward compactness, ordinary compactness, ordinary continuity, $\delta$-ward continuity, and slowly oscillating continuity.Comment: 16 pages. arXiv admin note: text overlap with arXiv:1005.494

Continuous selections of multivalued mappings

This survey covers in our opinion the most important results in the theory of continuous selections of multivalued mappings (approximately) from 2002 through 2012. It extends and continues our previous such survey which appeared in Recent Progress in General Topology, II, which was published in 2002. In comparison, our present survey considers more restricted and specific areas of mathematics. Note that we do not consider the theory of selectors (i.e. continuous choices of elements from subsets of topological spaces) since this topics is covered by another survey in this volume

Study of covering properties in fuzzy topology

This work is devoted to the study of covering properties both in L-fuzzy topological spaces and in smooth L-fuzzy topological spaces , that is the fuzzy spaces in Sostak's sense, where L is a fuzzy lattice . Based on the satisfactory theory of L-fuzzy compactness build up by Warner, McLean and Kudri, good definitions of feeble compactness and P-closedness are introduced and studied. A unification theory for good L-fuzzy covering axioms is provided. Following the lines of L-fuzzy compactness, we suggest two kinds of L-fuzzy relative compactness as in general topology, study some of their properties and prove that these notions are good extensions of the corresponding ordinary versions. We also present L-fuzzy versions of R-compactness , weak compactness and 0-rigidity and discuss some of their properties. By introducing 'a-Scott continuous functions', a 'goodness of extension' criterion for smooth fuzzy topological properties is established. We propose a good definition of compactness, which we call 'smooth compactness' in smooth L-fuzzy topological spaces. Smooth compactness turns out to be an extension of L-fuzzy compactness to smooth L-fuzzy topological spaces. We study some properties of smooth compactness and obtain different characterizations. As an extension of the fuzzy Hausdorffness defined by Warner and McLean, 'smooth Hausdorffness' is introduced in smooth L-fuzzy topological spaces. Good definitions of smooth countable compactness, smooth Lindelofness and smooth local compactness are introduced and some of their properties studied

Transfunctions and Other Topics in Measure Theory

Measures are versatile objects which can represent how populations or supplies are distributed within a given space by assigning sizes to subregions (or subsets) of that space. To model how populations or supplies are shifted from one configuration to another, it is natural to use functions between measures, called transfunctions. Any measurable function can be identified with its push-forward transfunction. Other transfunctions exist such as convolution operators. In this manner, transfunctions are treated as generalized functions. This dissertation serves to build the theory of transfunctions and their connections to other mathematical fields. Transfunctions that identify with continuous or measurable push-forward operators are characterized, and transfunctions that map between measures concentrated in small balls -- called localized transfunctions -- can be spatially approximated with measurable functions or with continuous functions (depending on the setting). Some localized transfunctions have fat graphs in the product space and fat continuous graphs are necessarily formed by localized transfunctions. Any Markov transfunction -- a transfunction that is linear, variation-continuous, total-measure-preserving and positive -- corresponds to a family of Markov operators and a family of plans (indexed by their marginals) such that all objects have the same instructions of transportation between input and output marginals. An example of a Markov transfunction is a push-forward transfunction. In two settings (continuous and measurable), the definition and existence of adjoints of linear transfunctions are formed and simple transfunctions are implemented to approximate linear weakly-continuous transfunctions in the weak sense. Simple Markov transfunctions can be used both to approximate the optimal cost between two marginals with respect to a cost function and to approximate Markov transfunctions in the weak sense. These results suggest implementing future research to find more applications of transfunctions to optimal transport theory. Transfunction theory may have potential applications in mathematical biology. Several models are proposed for future research with an emphasis on local spatial factors that affect survivorship, reproducibility and other features. One model of tree population dynamics (without local factors) is presented with basic analysis. Some future directions include the use of multiple numerical implementations through software programs

L-fuzzy compactness and related concepts

The compactness defined by Warner and McLean is extended to arbitrary L-fuzzy sets where L is a fuzzy lattice, i.e., a completely distributive lattice with an order reversing involution. It is shown that with our compactness we can build up a satisfactory theory. The different definitions of compactness in L-fuzzy topological spaces are stated and other characterizations of some of these notions are obtained. We also study their goodness and establish the inter-relations between the compactnesses which are good extensions. Good definitions of L-fuzzy regularity and normality are proposed. Following the lines of our compactness we suggest two definitions of L-fuzzy local compactness that are good extensions of the respective ordinary versions. A comparison between them is presented and some of their properties studied. A one point compactification is also obtained. By introducing a new definition of a locally finite family of L-fuzzy sets and combining it with our definition of compactness, we propose an L-fuzzy paracompactness and study some of its properties. Good definitions of L-fuzzy countable and sequential compactness and the Lindelof property are introduced and studied. We also present, in L-fuzzy topological spaces, good extensions of S-closedness and RS-compactness. Some of their properties are examined. Good L-fuzzy versions of almost compactness, near compactness and a strong compactness are put forward and studied. A comparison between these compactness related concepts is also presented