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Discovering the compactness properties in generalized-type metric spaces opens up a fascinating area of research. The present study tries to develop a theoretical framework for compactness with key properties in the recently developed interval metric space. This work begins with explaining the covers and open covers to define compact interval metric spaces and their main features. Next, a similar definition of compactness using the finite intersection property is introduced. Then, the famous Heine–Borel theorem for compactness is extended in the case of interval metric spaces. Also, the concepts of sequential-type compactness and Bolzano–Weierstrass (BW)-type compactness for interval metric spaces are introduced with their equivalency relationship. Finally, the notion of total boundedness in interval metric spaces and its connection with compactness is introduced, providing new insights into these mathematical concepts.

In this paper, we have generalized the interval-valued fuzzy metric 𝑀(𝑥, 𝑦, 𝑡) by allowing it to take the positive interval number instead of ordinary positive real number. In our case, both ' ' and the grade value '𝑀(𝑥, 𝑦, 𝑡)' are interval numbers.The underlying topology of this generalized interval-valued fuzzy metric (𝐺𝐼𝑉𝐹 metric) is studied. Two celebrated fixed point theorems of Banach and Edelstein are extended in this space. Also the problem related to image filter processing is studied.

Let (X,d)(X,d) be a metric space, and f:X→Xf:X\\rightarrow X be a continuous map. In this paper we prove that if R(f)R(f) is compact, and ω(x,f)≠∅\\omega (x,f)\\not =\\emptyset for all x∈Xx\\in X, then ff is equicontinuous if and only if there exist a pointwise recurrent isometric homeomorphism hh and a non-expanding map gg that is pointwise convergent to a fixed point v0v_{0} such that ff is uniformly conjugate to a subsystem (h×g)|S(h\\times g)|S of the product map h×gh\\times g. In addition, we give some still simpler necessary and sufficient conditions of equicontinuous graph maps.

This paper is concerned with maximum likelihood estimates for a large class of families of unimodal densities. The existence of measurable maximum likelihood estimates and the consistency of asymptotic maximum likelihood estimates are proved. By counterexamples it is shown that the conditions which are sufficient for consistency cannot be removed without compensation.