Analyzing the local Lindel¨of proper function and the local proper

function of deep learning in bitopological spaces

Ali A. Atoom 1,∗ , Hamza Qoqazeh2, Eman Hussein3, Anas Owledat4

1Department of Mathematics, Ajloun National University, P.O. Box 43, Ajloun, 26810, Jordan

2Department of Mathematics, Faculty of Science and Information Technology Irbid National University, P.O.

Box 2600, Irbid 21110, Jordan

3Department of Mathematics, Amman Arab University, P.O. Box 24 Amman, 11953, Jordan

4Ministry of Education, Amman, Jordan

Emails: aliatoom@anu.edu.jo; hhaaqq983@gmail.com; e.hussein@aau.edu.jo; emanbasssam@gmail.com

Abstract

It is essential to create new mathematical strategies to deal with everyday problems since they require a lot of

data and ambiguity. The best tool for doing this is proper functions, which are the most common mathematical

technique. In order to generate suitable functions, we investigate several set operators. A connection between

symmetry and certain types of proper functions and their classical topologies can be made. As a result of this

symmetry, we can examine the traits and behaviors of traditional topological notions through settings, and

vice versa. We describe a new class of proper functions in this paper and launch a preliminary investigation

into them. These functions are referred to as pairwise local proper functions and pairwise local Lindel¨of

proper functions in bitopological spaces. In general topology, we also establish the connection between this

new class of proper functions and other classes of generalized functions already in existence. Regarding the

new ideas, a number of relationships, necessary and sufficient conditions, examples and counter-examples are

provided. In addition, a different argument for the pairwise regularity of a pairwise Hausdorff and pairwise

locally compact bitopological space is presented. As part of this research, we also look at the images and

inverse images of specific bitopological features under these functions. A few product theorems pertaining to

these concepts were finally discovered.

Keywords: Bitopological spaces; Pairwise locally compact; Pairwise local lindel¨of; Pairwise proper function;

Pairwise locally proper functions; Pairwise local Lindel¨o proper functions