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In this article, we mainly formalize in Mizar [2] the equivalence among a few compactness definitions of metric spaces, norm spaces, and the real line. In the first section, we formalized general topological properties of metric spaces. We discussed openness and closedness of subsets in metric spaces in terms of convergence of element sequences. In the second section, we firstly formalize the definition of sequentially compact, and then discuss the equivalence of compactness, countable compactness,...
For a polish space M and a Banach space E let B1 (M, E)
be the space of first Baire class functions from M to E, endowed with the
pointwise weak topology. We study the compact subsets of B1 (M, E) and
show that the fundamental results proved by Rosenthal, Bourgain, Fremlin,
Talagrand and Godefroy, in case E = R, also hold true in the general
case. For instance: a subset of B1 (M, E) is compact iff it is sequentially
(resp. countably) compact, the convex hull of a compact bounded subset of
B1 (M,...
We characterize exactly the compactness properties of the product of κ copies of the space ω with the discrete topology. The characterization involves uniform ultrafilters, infinitary languages, and the existence of nonstandard elements in elementary extensions. We also have results involving products of possibly uncountable regular cardinals.
In this paper we first introduce the concept of compatible mappings of type (B) and compare these mappings with compatible mappings and compatible mappings of type (A) in Saks spaces. In the sequel, we derive some relations between these mappings. Secondly, we prove a coincidence point theorem and common fixed point theorem for compatible mappings of type (B) in Saks spaces.
The object of this paper is to establish a unique common fixed point theorem for six self-mappings satisfying a generalized contractive condition through compatibility of type and weak compatibility in a fuzzy metric space. It significantly generalizes the result of Singh and Jain [The Journal of Fuzzy Mathematics and Sharma [Fuzzy Sets and Systems . An example has been constructed in support of our main result. All the results presented in this paper are new.
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