The theories of metric spaces and fuzzy are crucial topics in mathematics. 
 Compactness is one the most important fundamental properties that have been widely used Functional Analysis. In this paper, definition compact soft space introduced some its theorems investigated. Also, sequentially locally defined relationships between them studied. Moreover, each previous two concepts several ot...

In this paper, for a given sequentially Yoneda-complete T1 quasi-metric space (X, d), the domain theoretic models of the hyperspace K0(X) of nonempty compact subsets of (X, d) are studied. To this end, the ω-Plotkin domain of the space of formal balls BX, denoted by CBX is considered. This domain is given as the chain completion of the set of all finite subsets of BX with respect to the Egli-Mi...

We prove a general principle satisfied by weakly precompact sets of Lipschitz-free spaces. By this principle, certain infinite dimensional phenomena in spaces over metric may be reduced to the same free their compact subsets. As easy consequences we derive several new and some known results. The main results are: $\mathcal F(X)$ is sequentially complete for every superreflexive Banach space $X$...

We introduce sequentially Sr modules over a commutative graded ring and sequentially Sr simplicial complexes. This generalizes two properties for modules and simplicial complexes: being sequentially Cohen-Macaulay, and satisfying Serre’s condition Sr . In analogy with the sequentially CohenMacaulay property, we show that a simplicial complex is sequentially Sr if and only if its pure i-skeleton...

Let X be a Hausdorff quotient of standard space (that is locally compact separable metric space). It shown that the following are equivalent: (i) image an irreducible map from space; (ii) has sequentially dense subset satisfying two technical conditions involving double sequences; (iii) whenever q : Y → , restriction ⁎ | V onto (where pure derived and closure set singleton fibres ). The proof u...

In this paper, we define several ideal versions of Cauchy sequences and completeness in quasimetric spaces. Some examples are constructed to clarify their relationships. We also show that: (1) if a quasi-metric space (X, ?) is I-sequentially complete, for each decreasing sequence {Fn} nonempty I-closed sets with diam{Fn} ? 0 as n ?, then ?n?N Fn single-point set; (2) let I be P-ideal, every pre...

Let A and B be closed, convex bounded subsets of a weakly sequentially complete Banach space E which both are not compact. Then there is linear form x0⁎∈E⁎ does attain its supremum on B. In particular, given any subset A⊂E, if every x⁎∈E⁎ either attains or infimum A, then relatively The same happens for finite family but compact subsets. result only remains true in arbitrary assuming, two σ(E⁎⁎...