I-favorable spaces:Revisited

International Journal of Engineering and Advanced Technology, 2019

this paper investigates the properties of conver- gence and sequentially compact for the generalized concepts –I-convergent and I -sequentially compact. We extend Cartesian product properties of convergence to I-convergence. Further, we develop some properties of -sequential compactness of a topologicalspace.

Topology and its Applications, 1995

We show that every quasi-metric space is isomorphic to a subspace of the hyperspace of a suitable metric space, endowed with the Hausdorff quasi-metric. Therefore a topological space is quasi-metrizable if and only if it can be embedded in a Hausdorff quasi-metric hyperspace.

Transactions of the American Mathematical Society, 1982

In this paper, we obtain results of the following type: if /: X-» Y is a closed map and X is some "nice" space, and Y2 is a &-space or has countable tightness, then the boundary of the inverse image of each point of Y is "small" in some sense, e.g., Lindelöf or «¿¡-compact. We then apply these results to more special cases. Most of these applications combine the "smallness" of the boundaries of the point-inverses obtained from the earlier results with "nice" properties of the domain to yield "nice" properties on the range. Introduction. Recall the following theorem due to Morita and Hanai [14] and Stone [17]. Theorem. // /: A-» Y is closed and X is metrizable, then the following are equivalent. (a) Y is first countable; (b) For each y E Y,df~x(y) is compact; (c) Y is metrizable. The (c) =» (b) part is due to Vaïnsteïn [22]. But even the (a) => (b) part holds under much more general conditions: Michael [7] showed (b) holds if A is paracompact, and Y is locally compact or first-countable. Note that the assumptions on Y in Michael's theorem could not be weakened to "Fis a ¿-space" or "F has countable tightness": the map identifying the limit points of a topological sum of k convergent sequences is a closed map from a metrizable space A to a Fréchet space Y, and | d/"'(y) | = k for some y E Y. In this paper, we show that the situation is different if we require Y2 to be a ¿-space or have countable tightness. (Recall that the square of a ¿-space or a space of countable tightness need not have the same property.) We will usually not be able to show that the boundaries of point-inverses are compact, but we will often (depending upon conditions imposed on A or Y) he able to show that they are "small" in some sense, e.g., Lindelöf or u,-compact. In the second section, we apply general results of this type to more special cases, often combining the "smallness" of the boundaries of point-inverses with "nice" properties of A to obtain "nice" properties of Y. We mention the following earlier result of the second author [21] which is related to this topic.

International Journal of Applied Mathematical Research, 2012

Quasi metrics have been used in several places in the literature on domain theory and the formal semantics of programming languages [1], [3]. In this paper we introduce the concept of generalized quasi metric(=gq) space and establish some fixed point theorems in gq metric spaces.

2015

The aim of this work is to extend interesting results on the metrizability of cone metric spaces as it appears in the literature. In this paper we appeal to quasiuniformities and uniformities to prove that a quasicone metric space is qausimetrizable, and from our results we will deduce that every cone metric space is metrizable; our approach is more on bitopological and topological properties and differs from the one used by the papers mentioned above but affirms some of their results.

Journal of Mathematics and Computer Science, 2016

In this paper, we prove some interesting results using forward and backward convergence in quasi cone metric spaces. We study forward and backward sequential compactness, sequential countably compactness, and sequential continuity property in quasi cone metric spaces and give some interesting results.

Topology and its Applications, 1993

Arhangel'skii, A.V. and D.N. Stavrova, On a common generalization of k-spaces and spaces with countable tightness, Topology and its Applications 51 (1993) 261-268. Several generalizations of tightness (such that in the countable case it could also serve as a generalization of k-spaces) are defined and studied. Cases in which the new invariants coincide with the usual tightness are obtained.

We present a survey of fixed point results in generalized metric spaces (g.m.s.) in the sense of Branciari , A fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces, Publ. Math. Debrecen, 57, 31-37]. Since it may happen that the topology of such space is not Hausdorff, several authors added Hausdorfness (or some other condition) as an additional assumption in order to obtain their results. We show here that such assumptions are usually superfluous. Finally, we state some open questions on the topic.

2013

Quaestiones Mathematicae

We prove versions of Ekeland, Takahashi and Caristi principles in preordered quasi-metric spaces, the equivalence between these principles, as well as their equivalence to some completeness results for the underlying quasi-metric space. These extend the results proved in S. Cobzaş, Topology Appl. 265 (2019), 106831, 22, for quasi-metric spaces. The key tools are Picard sequences for some special set-valued mappings on a preordered quasi-metric space X, defined in terms of the preorder and of a function ϕ on X.