volume-1.tex

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\hypersetup{pdftitle={General Topology as Ordered Semigroup Actions. Book 1},
 pdfauthor={Victor Porton},
 pdfsubject={general topology},
 pdfkeywords={discontinuous analysis,discontinuous calculus,algebraic general topology,quasi-uniform spaces,generalizations of proximity spaces,generalizations of nearness spaces,generalizations of uniform spaces}}
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\begin{document}

\noindent
\href{https://teachsector.com/limit/}{\includegraphics[scale=0.3]{img/Teach-Sector.jpeg}}

\noindent
Ad: Study \href{https://teachsector.com/limit/}{discontinuous analysis}, an enhanced
calculus in which every function is both differentiable and integrable.

\noindent
Ad: \href{https://teachsector.com/dforpython/}{World-best general purpose programming language}.
You won't like Python anymore.

\noindent
Ad: \href{https://science-dao.org}{Donate for science.}

\title{General Topology as Ordered Semigroup Actions. Book 1}


\author{Victor Porton}


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


\urladdr{\href{https://math.portonvictor.org}{https://math.portonvictor.org}}


\date{\today}


\thanks{\noun{Todd Trimble}, \noun{Andreas Blass}, \noun{Robert Martin
Solovay}, \noun{Niels Diepeveen}, and others (mentioned below) have proved some theorems which are now in this book.}
\begin{abstract}
\textbf{Introduced several new axiomatic systems, that are not less general than group theory, and discovered discontinuous analysis.
See~\cite{important} for an explanation why this theory is super-important.}

In this work I introduce and study in details the concepts of funcoids
which generalize proximity spaces and reloids which generalize uniform
spaces, and generalizations thereof. The concept of funcoid is generalized
concept of proximity, the concept of reloid is cleared from superfluous
details (generalized) concept of uniformity. 

Also funcoids and reloids are generalizations of binary relations
whose domains and ranges are filters (instead of sets). Also funcoids
and reloids can be considered as a generalization of (oriented) graphs,
this provides us with a common generalization of calculus and discrete
mathematics.

I consider (generalized) limit of arbitrary (discontinuous) function, defined in terms of funcoids.
Definition of generalized limit makes it obvious to define such things as derivative of an arbitrary function, integral of an arbitrary function, etc. It is given a definition of non-dif\-fe\-re\-nti\-able solution of a (partial) differential equation. It's raised the question how do such solutions ``look like'' starting a possible big future research program.

The generalized solution of one simple example differential equation is also considered.

The generalized derivatives and integrals are linear operators. For example $\int_a^b f(x)dx - \int_a^b f(x)dx = 0$ is defined and true for \emph{every} function.

The concept of continuity is defined by an algebraic formula (instead
of old messy epsilon-delta notation) for arbitrary morphisms (including
funcoids and reloids) of a partially ordered category. In one formula
continuity, proximity continuity, and uniform continuity are generalized.

Also I define connectedness for funcoids and reloids.

Then I consider generalizations of funcoids: pointfree funcoids and
generalization of pointfree funcoids: staroids and multifuncoids.
Also I define several kinds of products of funcoids and other morphisms.

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 semicategory 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.

Before going to topology, this book studies properties of co-brouwerian
lattices and filters.
\end{abstract}


\keywords{algebraic general topology, quasi-uniform spaces, generalizations
of proximity spaces, generalizations of nearness spaces, generalizations
of uniform spaces, limit, ordered semigroups, semigroup actions}


\subjclass[2010]{54J05, 54A05, 54D99, 54E05, 54E15, 54E17, 54E99}

\maketitle

\pagebreak

{
  \LARGE
This book was self-published under a free license by a person without scientific degree.
I can't republish it in a reputable publisher, because only degree holders can receive grants.

I discovered that PhDs want to build on only on works of other PhDs.
Thus by publishing it, I broke the desire of PhDs to participate in research of ordered semigroup actions.

Ordered semigroup actions may be left not researched, because no one wants to build on my research.
\emph{Oh sorry, I broke PhDs.} See~\cite{broke-science} for more information.
In other words, I did a scientific ``covery'': when I cover a research area from future attention, even
if rediscovered.

Discovery of ordered semigroup actions is super-important by itself, but it is made finally invaluable
by their missingness in scientific databases (like billion dollars for the missing cup of water in
a wilderness).

Break this bond: Starts your own research based on this book. If you don't do, humanity lost ordered semigroup actions finally.

\noindent\makebox[\linewidth]{\rule{\textwidth}{0.4pt}}

  \Large
After noting that actions of ordered semigroups and discontinuous analysis are ``needed''
to nobody on the Earth I prayed ``God, take me to heaven alone.''
Get this book or go to the hell.

As a scientist, you should understand that ordered semigroup actions and Discontinuous Analysis
are critical for science and their non-publication draws back world economy.
So, you should agree with me, that \href{https://science-dao.org}{donation} for publication of
such discoveries may benefit you personally more than you spend, because of its multiplicative
or exponential effect on the world economy.
Non-donors go to the hell, because of being so much greedy that don't donate even for their own
benefit.

The Apocalypse's ``stamp on the forehead'' (somehow related to the number 666) is when
you by your forehead believe that degree (the stamp) is essential for doing science.
This stamp transforms science into a stupid ``beast'': Your academia cannot learn even
ordered semigroup actions.
}

\tableofcontents{}

\part{Introductory chapters}
\include{chap-intro}
\include{chap-common}
\include{chap-order-more}
\include{chap-rel}
\include{chap-filt}
\include{chap-common-top}

\part{Funcoids and reloids}
\include{chap-funcoids}
\include{chap-reloids}
\include{chap-relationships}
\include{chap-decomposition}
\include{chap-continuity}
\include{chap-connected}
\include{chap-bound}
\include{chap-filt-order}
\include{chap-fcd-counter}
\include{chap-funcoids-are-filters}
\include{chap-cofinite}
\include{chap-convergence}
\include{chap-unfixed}

\part{Pointfree funcoids and reloids}
\include{chap-pf-funcoids}
\include{chap-alt-bin}

\part{Staroids and multifuncoids}
\include{chap-disjoint}
\include{chap-multi}

\part{Algebra of general topology}

\include{chap-osgroups}

\part{Generalized limit}
% \chapter{Generalized limit}
Discontinuous analysis chapter removed.
Instead apply for this paid course:
\url{https://teachsector.com/limit/}

% \include{limit}

\part{Postface}
\include{chap-postface}
\include{chap-prize}
\include{chap-story}

\include{appendixes}

\printindex{}

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