volume-3.tex

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\hypersetup{pdftitle={Algebraic General Topology. Book 3: Algebra. Edition 3},
 pdfauthor={Victor Porton},
 pdfsubject={general topology, algebra},
 pdfkeywords={algebraic general topology,quasi-uniform spaces,generalizations of proximity spaces,generalizations of nearness spaces,generalizations of uniform spaces,generalizations of metric spaces,ordered semigroups,ordered monoids,abstract algebra,universal algebra}}

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\begin{document}

\title{Algebraic General Topology.\\Book 3: Algebra.\\Edition 3}

\author{Victor Porton}

\email{\href{mailto:porton@narod.ru}{porton@narod.ru}}

\urladdr{\href{http://www.mathematics21.org}{http://www.mathematics21.org}}

\date{\today}

\begin{abstract}
This is the world first study of \emph{ordered semigroup actions}.

I define \emph{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 \emph{interspaces}). I build basic general topology (continuity, limit, openness, closedness, hausdorffness, compactness, 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.

Was a spell laid onto Earth mathematicians not to find the most important structure in general topology until 2019?
\end{abstract}


\keywords{algebraic general topology, quasi-uniform spaces, generalizations of proximity spaces, generalizations of nearness spaces, generalizations
of uniform spaces, ordered semigroups, ordered monoids, abstract algebra, universal algebra}


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This is a draft.

It is a continuation of \cite{volume-1}.

You can read this text without any knowledge of algebraic general topology~(\cite{volume-1}). But to have some examples of how to apply this theory, you need to know what funcoids and reloids are and how funcoids are related with topological spaces.

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