On structured spaces and their properties

TL;DR

This paper introduces 'structured spaces', a new class of topological spaces that locally resemble algebraic structures, providing a framework to study spaces with local algebraic properties and establishing a duality with lattices.

Contribution

It defines structured spaces via the structure map, explores their properties, and proves a duality theorem linking these spaces with lattices, advancing the understanding of local algebraic structures in topology.

Findings

Structured spaces induce lattices under certain conditions.

Lattices can generate structured spaces satisfying specific hypotheses.

Connections between structured spaces and connected spaces are established.

Abstract

In this paper we introduce a new kind of topological space, called 'structured space', which locally resembles various kinds of algebraic structures. This can be useful, for instance, to locally study a space that cannot be globally endowed with an algebraic structure using tools from algebra. The definition of these spaces will be made more precise via one of our main result, which involves the 'structure map'. This will also lead us to a rigorous and unambiguous definition of algebraic structure. After showing some examples which naturally arise in this context, we study various properties and develop some theory for these new spaces; in particular, we consider partitions (with respect to some measure $\mu$). We then prove one of the most important Theorem of this paper (Theorem 4.1), which states that every structured space, under some assumptions, induces a lattice, and conversely every lattice induces a structured space satisfing such hypothesis. We conclude with some relations with connected spaces.