Abstract
The structure of topological spaces is analysed here through the lenses of fibrous preorders. Each topological space has an associated fibrous preorder and those fibrous preorders which return a topological space are called spatial. A special class of spatial fibrous preorders consisting of an interconnected family of preorders indexed by a unitary magma is called Cartesian and is studied here. Topological spaces that are obtained from those fibrous preorders, with a unitary magma I, are called I-Cartesian and are characterized. The characterization reveals a hidden structure on such spaces. Several other characterizations are obtained, and special attention is drawn to the case of a monoid equipped with a topology. A wide range of examples is provided, as well as general procedures to obtain topologies from other data types such as groups and their actions. Metric spaces and normed spaces are considered as well.
Highlights
A Cartesian spatial fibrous preorder, indexed by the unitary magma ( I, ·, 1), is a system ( X, (≤i )i∈ I, (∂i )i∈ I ) where X is a set and for every i ∈ I, ≤i is a binary relation on X, whereas ∂i is a partial map X × X → I, which is defined for all pairs ( x, y) such that x ≤i y
It is clear that when a Cartesian spatial fibrous preorder is obtained from a metric space, its induced topology is the same as the usual topology generated by the metric
The notion of Cartesian spatial fibrous preorder has been introduced as a system ( X, (≤i )i∈ I, (∂i )i∈ I ) indexed over a unitary magma, I, and satisfying conditions (C1), (C2), and (C3)
Summary
If X is a finite set, as is well known [22], a topology on X is nothing but a preorder, which is a reflexive and transitive relation Should this not be a simple observation that would follow from the definition of a topological space?. A topological group is presented as a group equipped with an arbitrary topology (which is required to be compatible with the group operation), and it is clear that such topologies must be simpler than arbitrary ones This is mainly because the simplification of the structure of those topologies is not apparent from its definition.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
