# Pairwise Semiregular Properties on Generalized Pairwise Lindelof Spaces

## TL;DR

This paper investigates the concept of pairwise semiregular properties in bitopological spaces, establishing which Lindel"{o}f-related properties are preserved under pairwise semiregularization.

## Contribution

It introduces the notion of pairwise semiregular properties and characterizes which Lindel"{o}f-type properties are preserved under this process.

## Key findings

- (i,j)-almost Lindel"{o}f, pairwise almost Lindel"{o}f, (i,j)-weakly Lindel"{o}f, and pairwise weakly Lindel"{o}f are pairwise semiregular properties.
- Certain types of pairwise Lindel"{o}f spaces are not pairwise semiregular properties.

## Abstract

Let $\left( X,\tau _{1},\tau _{2}\right) $ be a bitopological space and $% \left( X,\tau _{\left( 1,2\right) }^{s},\tau _{\left( 2,1\right) }^{s}\right) $ its pairwise semiregularization. Then a bitopological property $\mathcal{P}$\ is called pairwise semiregular provided that $\left( X,\tau _{1},\tau _{2}\right) $\ has the property $\mathcal{P}$\ if and only if $\left( X,\tau _{\left( 1,2\right) }^{s},\tau _{\left( 2,1\right) }^{s}\right) $\ has the same property. In this work we study pairwise semiregular property of $\left( i,j\right) $-nearly Lindel\"{o}f, pairwise nearly Lindel\"{o}f, $\left( i,j\right) $-almost Lindel\"{o}f, pairwise almost Lindel\"{o}f, $\left( i,j\right) $-weakly Lindel\"{o}f and pairwise weakly Lindel\"{o}f spaces. We prove that $\left( i,j\right) $-almost Lindel% \"{o}f, pairwise almost Lindel\"{o}f, $\left( i,j\right) $-weakly Lindel\"{o}% f and pairwise weakly Lindel\"{o}f are pairwise semiregular properties, on the contrary of each type of pairwise Lindel\"{o}f space which are not pairwise semiregular properties.