# A space with a Lusin $\pi$-base whose square has no Lusin $\pi$-base

## TL;DR

This paper constructs a topological space with a Lusin π-base whose Cartesian square lacks such a base, highlighting a unique property of these spaces.

## Contribution

It presents the first example of a space with a Lusin π-base whose square does not have a Lusin π-base, revealing new insights into their structural properties.

## Key findings

- Constructed a space with a Lusin π-base but its square lacks one.
- Demonstrated the non-preservation of Lusin π-base property under Cartesian square formation.
- Provided a counterexample to previous assumptions about Lusin π-bases.

## Abstract

We construct a space ${X}$ that has a Lusin $\pi$-base and such that ${X}^{2}$ has no Lusin $\pi$-base.