I define space as an element of an ordered semigroup action, that is a semigroup action conforming to a partial order. Topological spaces, uniform spaces, proximity spaces, directed graphs, metric spaces, etc. all are spaces. It can be further generalized to ordered precategory actions (that I call interspaces). I build basic general topology (continuity, limit, openness, closedness, hausdorffness, compactness, connectedness, etc.) in an arbitrary space. Now general topology is an algebraic theory.
For example, my generalized continuous function are: continuous function for topological spaces, proximally continuous functions for proximity spaces, uniformly continuous functions for uniform spaces, contractions for metric spaces, discretely continuous functions for (directed) graphs.
Google search for both “”ordered semigroup action”” and “”action of ordered semigroup”” showed nothing. Was a spell laid onto Earth mathematicians not to find the most important structure in general topology until 2019?
This book was developed with the caring and concerned adult in mind and is a one-stop for anyone who would like to help a student establish abstract math thinking. They will become adept at the use of math proof strategies throughout their development from birth.
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Elementary general topology theory concentrates on results with a specific geometric meaning, with little algebraic formalism, thus providing proof that some highly nontrivial results are obtained through the procedure.
Important applications become immediately evident without the need for a broad formal.
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