# Combinatorial properties on nodec countable spaces with analytic topology

## TL;DR

This paper explores specialized topological spaces with analytic topology, constructing countable nodec regular spaces that are not selectively separable and lack the $q^+$ property, advancing understanding of their combinatorial and topological properties.

## Contribution

It introduces new variations of product topology on clopen sets to construct specific countable nodec regular spaces with unique combinatorial features.

## Key findings

- Constructed countable nodec regular spaces with analytic topology
- Spaces are not selectively separable
- Spaces do not satisfy the $q^+$ principle

## Abstract

We study some variations of the product topology on families of clopen subsets of $2^{\mathbb{N}}\times\mathbb{N}$ in order to construct countable nodec regular spaces (i.e. in which every nowhere dense set is closed) with analytic topology which in addition are not selectively separable and do not satisfy the combinatorial principle $q^+$.