On the $1-1$ sum of two Borel sets

The hierarchy of ω 1 - Borel sets 1 The hierarchy of ω 1 - Borel sets Arnold

We consider the ω1-Borel subsets of the reals in models of ZFC. This is the smallest family of sets containing the open subsets of the 2 and closed under ω1 intersections and ω1 unions. We show that Martin’s Axiom implies that the hierarchy of ω1-Borel sets has length ω2. We prove that in the Cohen real model the length of this hierarchy is at least ω1 but no more than ω1 + 1. Some authors have...

Homogeneous Borel Sets of Ambiguous Class Two

We describe and characterize all homogeneous subsets of the Cantor set which are both an F„s and a GSo; it turns out that there are wx such spaces.

Group ${1, -1, i, -i}$ Cordial Labeling of sum of $C_n$ and $K_m$ for some $m$

Let G be a (p,q) graph and A be a group. We denote the order of an element $a in A $ by $o(a).$  Let $ f:V(G)rightarrow A$ be a function. For each edge $uv$ assign the label 1 if $(o(f(u)),o(f(v)))=1 $or $0$ otherwise. $f$ is called a group A Cordial labeling if $|v_f(a)-v_f(b)| leq 1$ and $|e_f(0)- e_f(1)|leq 1$, where $v_f(x)$ and $e_f(n)$ respectively denote the number of vertices labelled w...

A Course on Borel Sets

Borel - Cantelli Lemma 1

The notation and terminology used here have been introduced in the following papers: [17], [3], [4], [8], [13], [1], [2], [5], [15], [14], [21], [9], [12], [11], [16], [6], [20], [19], and [18]. For simplicity, we adopt the following rules: O1 is a non empty set, S1 is a σ-field of subsets of O1, P1 is a probability on S1, A is a sequence of subsets of S1, and n is an element of N. Let D be a s...