We show that the category of convergence approach SPACES is a simultaneously re ective and core ective subcategory of the category of latticevalued limit SPACES. Further we study the preservation of diagonal conditions, which characterize approach SPACES. It is shown that the category of preapproach SPACES is a simultaneously re ective and core ective subcategory of the category of lattice-valued pretopological SPACES and that the category of approach SPACES is a core ective subcategory of a category of lattice-valued topological convergence SPACES.

Trace of quotient SPACES is usually seen wherever there is a study of a linear structure. In linear SPACES, we use subSPACES and their corresponding equivalence relation to define quotient SPACES. With the same method, in this paper, we present two generalized structures of quotient space that are defined on quasilinear SPACES. One of them is a quasilinear space and the other is a linear space. After that, we try to introduce norms on certain states of these SPACES and examin some properties of them. We will also provide examples for better understanding throughout the process.

In this paper, the concept of a pseudosymmetric space which is a natural generalization of the concept of a symmetric space is defined. All basic concepts such as the Luxemburg representation theorem, the Boyd indices, the fundamental function and its properties, Calderon’, s theorem, etc., is transferred over the pseudosymmetric case. Examples are given for pseudosymmetric SPACES. The quasi-symmetric SPACES expand the scope of the application of symmetric space results.

For a class of fuzzy metric SPACES (in the sense of George and Veeramani) with an H-type t-norm, we present a method to construct a metric on a fuzzy metric space. The induced metric space shares many important properties with the given fuzzy metric space. Specifically, they generate the same topology, and have the same completeness. Our results can give the constructive proofs to some problems for fuzzy metric SPACES in the literature, which are shown by examples in this paper.

In this paper, we discuss the inheritance of strict convexity, uniform convexity and local uniform convexity by the quotient SPACES of metric linear SPACES. We also show that, as in the case of normed linear SPACES, completeness is a three-space property in metric linear SPACES as well.

In this paper we introduce the class of grand Lebesgue SPACES associated to a Banach function space $X$ by replacing the role of the $L^1$-norm by the norm $\|\cdot\|_X$ in the classical construction of the generalized grand Lebesgue SPACES. Also, we study the inclusion property of these SPACES.

The L-fuzzy approximation operator associated with an L-fuzzy approximation space (X, R) turns out to be a saturated L-fuzzy closure (interior) operator on a set X precisely when the relation R is reflexive and transitive. We investigate the relations between L-fuzzy approximation SPACES and L-(fuzzy) topological SPACES.