proof:(Cantor Intersection Theorem) 第一证明这个交集closed,第二证明这个交集非空 S=⋂k=1∞Qk (closed非常好证,一堆闭集的交集还是闭集) 证明非空:第一步我们得先意识到, Qk内一定有无穷多个点(如果有穷的话,这个就是trivial case,一环套一环到最后的那个集合里面一定会剩下东西,然后就可以证明非空)...
Lakshmi Kanta DeySumit Som
We establish an extension of Cantor’s intersection theorem for a \({K}\)-metric space (\({X, d}\)), where \({d}\) is a generalized metric taking values in a solid cone \({K}\) in a Banach space \({E}\). This generalizes a recent result of Alnafei, Radenović and Shahza...
In this paper, for the first time, we establish Cantor's intersection theorem and Baire category theorem in 2-metric spaces. As a departure from normal practice we then apply Cantor's theorem to establish some fixed point theorems in such spaces....
Cantor's best known proof is his technique ofCantorian Diagonalization, a method useful to prove that thereal numbersare larger incardinalitythan theintegers. He is also known for his Cantor Set[8]andCantor Intersection Theorem. Contents
The Cantor's Nested Interval Theorem, also known as the Bolzano's Theorem, is a fundamental result in mathematics that deals with the concept of closed intervals. It states that if we have a sequence of closed intervals, each contained within the previous one, then the intersection of all th...
the unit interval [0, 1], let q = 2s +1 for s = 1, 2, 3, ···; then let us consider the affine maps: T s,i (x) = ( 1 q )x +( 2i q ) i = 0, 1, ··· , s Now let I s,0 = [0, 1] and let n = 1, 2, ···, then we define I s,n ...
A CANTOR TYPE INTERSECTION THEOREM FOR SUPERREFLEXIVE BANACH SPACES AND FIXED POINTS OF ALMOST AFFINE MAPPINGS We obtain the following Cantor type intersection t Jachymski,Jacek - 《Journal of Nonlinear & Convex Analysis》 被引...
was long a concern of Cantor's.[44] He directly addressed this intersection between these disciplines in the introduction to his Grundlagen einer allgemeinen Mannigfaltigkeitslehre, where he stressed the connection between his view of the infinite and the philosophical one.[45] To Cantor, his ma...
In particular, the set has this property (compare Theorem 2.2). Moreover, is a finite union of closed intervals if and only if for sufficiently large n’s (see Theorem 2.3). A main goal of our paper is to find conditions which imply that the difference set is a Cantorval. We examine...