On structured spaces and their properties

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.

arXiv comment30 pages; added new examples and references

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Cited by in corpus (5)

• Two cohomology theories for structured spaces
• (Co)homology theories for structured spaces arising from their corresponding poset
• A (co)homology theory for some preordered topological spaces
• Conical calculus on schemes and perfectoid spaces via stratification
• Structured spaces: categories, sheaves, bundles, schemes and cohomology theories