Bases and Refinements of Topologies 1
نویسنده
چکیده
In this article we present several logical schemes. The scheme FraenkelInvolution deals with a non empty set A , subsets B, C of A , and a unary functor F yielding an element of A , and states that: B = {F (a);a ranges over elements of A : a ∈ C} provided the following conditions are satisfied: • C = {F (a);a ranges over elements of A : a ∈ B}, and • For every element a of A holds F (F (a)) = a. The scheme FraenkelComplement1 deals with a non empty relational structure A , a family B of subsets of A , a set C , and a unary functor F yielding a subset of A , and states that: Bc = {F (a)c;a ranges over elements of A : a ∈ C} provided the following condition is met: • B = {F (a);a ranges over elements of A : a ∈ C}. The scheme FraenkelComplement2 deals with a non empty relational structure A , a family B of subsets of A , a set C , and a unary functor F yielding a subset of A , and states that: Bc = {F (a);a ranges over elements of A : a ∈ C} provided the following requirement is met: • B = {F (a)c;a ranges over elements of A : a ∈ C}. Next we state several propositions:
منابع مشابه
Refinements of the Density and ^-density Topologies
Given an arbitrary ideal f on the real numbers, two topologies are denned that are both finer than the ordinary topology. There are nonmeasurable, non-Baire sets that are open in all of these topologies, independent of f . This shows why the restriction to Baire sets is necessary in the usual definition of the ./"-density topology. It appears to be difficult to find such restrictions in the cas...
متن کاملOn generalized topologies arising from mappings
Given a mapping $f:Xto X$ we naturally associate to it a monotonic map $gg_f:exp Xto exp X$ from the power set of $X$ into itself, thus inducing a generalized topology on $X$. In this paper we investigate some properties of generalized topologies which are defined by such a procedure.
متن کاملPreorder Relators and Generalized Topologies
In this paper we investigate generalized topologies generated by a subbase of preorder relators and consider its application in the concept of the complement. We introduce the notion of principal generalized topologies obtained from the new type of open sets and study some of their important properties.
متن کاملPre-service Language Teachers’ Cognitions about Language Learning/teaching and Cognition Refinements through a Reflection-oriented Practicum
Pre-service teacher cognition literature abounds with the concept of change as well as proposals regarding desirable conceptual changes to be undertaken so that teacher candidates can set out their path of professional development with right beliefs. Yet, such changes would seem less likely to occur if the trainees’ thoughts, beliefs, and cognitions are not identified in the f...
متن کاملTopologies refining the Cantor topology on X
The space of one-sided infinite words plays a crucial rôle in several parts of Theoretical Computer Science. Usually, it is convenient to regard this space as a metric space, the Cantor-space. It turned out that for several purposes topologies other than the one of the Cantor-space are useful, e.g. for studying fragments of first-order logic over infinite words or for a topological characterisa...
متن کامل