Continuous functions between sets with operations

Arithmetic Operations on Functions from Sets into Functional Sets

In this paper x, X, X1, X2 are sets. Let Y be a functional set. The functor DOMS(Y ) is defined by: (Def. 1) DOMS(Y ) = ⋃ {dom f : f ranges over elements of Y }. Let us consider X. We say that X is complex-functions-membered if and only if: (Def. 2) If x ∈ X, then x is a complex-valued function. Let us consider X. We say that X is extended-real-functions-membered if and only if: (Def. 3) If x ∈...

Micro Tangent Sets of Continuous Functions

Motivated by the concept of tangent measures and by H. Fürstenberg’s definition of microsets of a compact set A we introduce micro tangent sets and central micro tangent sets of continuous functions. It turns out that the typical continuous function has a rich (universal) micro tangent set structure at many points. The Brownian motion, on the other hand, with probability one does not have graph...

K -Trivial Closed Sets and Continuous Functions

We investigate the notion of K-triviality for closed sets and continuous functions. Every K-trivial closed set contains a K-trivial real. There exists a K-trivial Π 1 class with no computable elements. For any K-trivial degree d, there is a K-trivial continuous function of degree d.

Γ-sets and Γ-continuous Functions

On Nonspecificity of fuzzy sets with continuous membership functions

A measure of nonspecificity is proposed for fuzzy sets with continuous membership functions defined, in general, on the n-dimensional Euclidean space R". The proposed measure is restricted to fuzzy sets that are convex. The measure is justified in terms of the usual axiomatic requirements.