Product Methods And G -Connectedness

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PRODUCT METHODS AND G-CONNECTEDNESS L. LIU and Z. PING∗ School of Mathematics and Physics, Ningde Normal University, Ningde, Fujian 352100, P. R. China e-mails: [email protected], [email protected] (Received April 18, 2019; revised May 6, 2020; accepted May 12, 2020)

Abstract. The generalized topology on the Cartesian product of sets can be defined by the generalized topology on the factors of the product. A G-method on a set can derive a generalized topology, which is called a G-generalized topology. On the one hand, we introduce the concept of product G-methods on sets which lead to a G-generalized topology on the Cartesian products that is different both from the Cs´ asz´ ar and the Eckhoff product of G-generalized topologies on the factors. On the other hand, we study the G-connectedness of the Cartesian products and prove that a G-connectedness determined by an almost pointwise method is countably multiplicative. The countably multiplicative property of sequentially connected spaces is extended.

1. Introduction Sequential convergence is a basic concept in mathematics [7]. As a generalization of convergence, a wide variety of convergence types can be discussed in topological spaces. For example, there are the convergence of filters, nets and set families, statistical convergence, ideal convergence, and so on. On this basis, we can define G-convergence and G-open sets on arbitrary sets and derive its G-generalized topology [11]. G-generalized topology is a kind of generalized topology, so the results of generalized topology can all be applied to G-generalized topology [3]. The product of generalized topologies is similar to the Tychonoff product in topological spaces [5]. Accordingly, we can introduce the generalized topology on the Cartesian products through the product of G-generalized topologies on the factors of the product. Since ∗ Corresponding

author. This research is supported by NSFC (No. 11801254) and Scientific Research and Innovation Team Project of Ningde Normal University (2017T01). Key words and phrases: generalized topology, γ-connectedness, G-method, product method, pointwise method, almost pointwise method. Mathematics Subject Classification: 54A20, 54C08, 54B15, 54D55, 40A05, 40C99.

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L. LIU LIU and and Z. Z. PING PING L.

the key of G-generalized topology is the G-methods, we are particularly concerned about the G-generalized topology derived from G-methods. It involves the product methods of G-methods. The following question is interesting: Does the G-generalized topology on the Cartesian products derived by this way coincide with the generalized topology generated by the product of G-generalized topology on the factors? As one class of homeomorphic properties, connectedness plays an important role in topology and related disciplines [7]. Recently, O. Mucuk and H. C ¸ akallı [17] introduced the finite product method of G-methods on topological groups, and proved that G-connectedness is finitely productive for regular G-methods. As we