This is even more idiosyncratic than other mathematical items and relates only VERY indirectctly to Maksakovsky via my personal interest in clarifying fundamental concepts while learning/refreshing basic mathematics needed for explicating and further developing theory of the capitalist cycle based on materialist dialectics in mathematics as well as economics.

Perhaps should just be in my personal project section but may help shed light on both the distractions and relevance of other items.

Some basic topology is obviously essential.

Proximity spaces (and related Nearness) give most “natural” definition of continuity:

Close sets have close images. Ie continous maps dont tear things apart. Corresponds to naive “draw curve without lifting pencil from paper” but precise and rigorous as a point (singleton subset) is near a set iff in closure of set and a set is close to another iff all its points are near.

This is covariant and intuitive unlike contravariant and obscure epsilontics.

Not usually taught, presumably for historical/hysterical reasons.

Should work well with transition from topology to uniformity to approach space (extended pseudoquasimetrics) to metrics to norms to inner products to geometric algebra. Hopefully also to general integral via inversion of differential forms from universal geometric calculus.

Also distance to set as an infimum or glb should fit well with infinitesimals (and approach spaces are used for convergence in measure and probability so presumably general integral).

I like closure axioms more than open sets or neighbourhoods. Mainly because closures also relevant far more widely (including, boundaries, convexity, galois connections, other adjoints, “dialectics”). In turn indepotent morphisms even more central and include closures.

Perhaps “extensive” axiom in closure operators not just a pun but some relation to extensionality. Injective (extensional )medial Auto-magma inseparable opposites are a commutative half group. Idempotent free or nearly rigid? Arithmos?

Attracted to boundary/frontier as reflecting both. Have paper on axiomatization of topological boundary but does not seem intuitive.

Convergence goes well with nets based on intuition from sequences but I prefer (equivalent) formulation with filters.

Separation axioms (T0 etc) are fundamental and easily expressed. Includes connectedness.

Not sure how compactness and local compactness is expressed.

Hopefully the centrality of ** R** and

**in linear algebra can be explained based on being the only connected locally compact topological fields in terms of continuous convergence spaces and duality of evaluation maps.**

*C*For constructive foundations separation is not just negation of nearness since without excluded middle continuity isn’t interesting.

Hopefully this can provide starting text:

Naimpally S.A., Peters J.F.-Topology with Applications_ Topological Spaces via Near and Far-World Scientific (2013).pdf

The principal aim of this book is to introduce topology and its many applications viewed within a framework that includes a consideration of compactness, completeness, continuity, filters, function spaces, grills, clusters and bunches, hyperspace topologies, initial and final structures, metric spaces, metrization, nets, proximal continuity, proximity spaces, separation axioms, and uniform spaces.

This book provides a complete framework for the study of topology with a variety of applications in science and engineering that include camouflage filters, classification, digital image processing, forgery detection, Hausdorff raster spaces, image analysis, microscopy, paleontology, pattern recognition, population dynamics, stem cell biology, topological psychology, and visual merchandising.

It is the first complete presentation on topology with applications considered in the context of proximity spaces, and the nearness and remoteness of sets of objects. A novel feature throughout this book is the use of near and far, discovered by F Riesz over 100 years ago. In addition, it is the first time that this form of topology is presented in the context of a number of new applications.

Readership: 3rd year undergraduate students, graduate students and researchers in topology; professional and practitioners who are interested in applying topology and its applications especially in science and engineering

@book{book:1122439,

title = {Topology with Applications: Topological Spaces via Near and Far},

author = {Naimpally S.A., Peters J.F.},

publisher = {World Scientific},

isbn = {978-981-4407-65-6},

year = {2013},

series = {},

edition = {1},

volume = {},

url = {http://gen.lib.rus.ec/book/index.php?md5=07c7235ab1630059841ba360111d7e47}

}

K.D. Joshi-Introduction to general topology-John Wiley & Sons (Asia) Pte Ltd (1983).djvu

@book{book:145179,

title = {Introduction to general topology},

author = {K.D. Joshi},

publisher = {John Wiley & Sons (Asia) Pte Ltd},

isbn = {0852264445,9780852264447},

year = {1983},

series = {},

edition = {},

volume = {},

url = {http://gen.lib.rus.ec/book/index.php?md5=6A9FD5B8B17E02BF2BB15ED807808150}

}

@article{Gregor-2008,

doi = {10.1016/j.hm.2008.01.001},

title = {The emergence of open sets, closed sets, and limit points in analysis and topology},

author = {Gregory H. Moore},

publisher = {Elsevier Science},

journal = {Historia Mathematica},

issnp = {0315-0860},

issne = {1090-249X},

year = {2008},

volume = {35},

issue = {3},

page = {220–241},

url = {http://gen.lib.rus.ec/scimag/index.php?s=10.1016/j.hm.2008.01.001},

}

@article{P-1974,

doi = {10.2307/2319561},

title = {Nearness–A Better Approach to Continuity and Limits},

author = {P. Cameron, J. G. Hocking and S. A. Naimpally},

publisher = {Mathematical Association of America},

journal = {American Mathematical Monthly},

issnp = {0002-9890},

issne = {1930-0972},

year = {1974},

month = {08-09},

volume = {81},

issue = {7},

page = {739–745},

url = {http://gen.lib.rus.ec/scimag/index.php?s=10.2307/2319561},

}

@article{Velleman-1997,

doi = {10.1080/00029890.1997.11990643},

title = {Characterizing Continuity},

author = {Velleman, Daniel J.},

publisher = {Mathematical Association of America},

journal = {American Mathematical Monthly},inbb

issnp = {0002-9890},

issne = {1930-0972},

year = {1997},

month = {04},

volume = {104},

issue = {4},

page = {318–322},

url = {http://gen.lib.rus.ec/scimag/index.php?s=10.1080/00029890.1997.11990643},

}

## Constructive

@article{L.S-2003,

doi = {10.1016/s0304-3975(02)00711-9},

title = {A constructive theory of point-set nearness},

author = {L.S. Vı̂ţă; D.S. Bridges},

publisher = {Elsevier Science},

journal = {Theoretical Computer Science},

issnp = {0304-3975},

year = {2003},

volume = {305},

issue = {1-3},

page = {473–489},

url = {http://gen.lib.rus.ec/scimag/index.php?s=10.1016/s0304-3975(02)00711-9},

}

@article{Dougla-2008,

doi = {10.1016/j.tcs.2008.06.019},

title = {Apartness, compactness and nearness},

author = {Douglas Bridges; Hajime Ishihara; Peter M. Schuster; Luminiţa Vîţă},

publisher = {Elsevier Science},

journal = {Theoretical Computer Science},

issnp = {0304-3975},

year = {2008},

volume = {405},

issue = {1-2},

page = {3–10},

url = {http://gen.lib.rus.ec/scimag/index.php?s=10.1016/j.tcs.2008.06.019},

}

## Advanced

(Cambridge Tracts in Mathematics 59) S. A. Naimpally, B. D. Warrack-Proximity Spaces-Cambridge University Press (2008).pdf

@book{book:825284,

title = {Proximity Spaces},

author = {S. A. Naimpally, B. D. Warrack},

publisher = {Cambridge University Press},

isbn = {0521091837,9780521091831,0521079357,9780521079358},

year = {2008},

series = {Cambridge Tracts in Mathematics 59},

edition = {1},

volume = {},

url = {http://gen.lib.rus.ec/book/index.php?md5=1aa4ccc0bd94413da539555345e68c98}

}

https://arxiv.org/abs/1001.1866

## Beyond Topologies, Part I

(Submitted on 12 Jan 2010)Arguments on the need, and usefulness, of going beyond the usual Hausdorff-Kuratowski-Bourbaki, or in short, HKB concept of topology are presented. The motivation comes, among others, from well known {\it topological type processes}, or in short TTP-s, in the theories of Measure, Integration and Ordered Spaces. These TTP-s, as shown by the classical characterization given by the {\it four Moore-Smith conditions}, can {\it no longer} be incorporated within the usual HKB topologies. One of the most successful recent ways to go beyond HKB topologies is that developed in Beattie & Butzmann. It is shown in this work how that extended concept of topology is a {\it particular} case of the earlier one suggested and used by the first author in the study of generalized solutions of large classes of nonlinear partial differential equations.

https://arxiv.org/abs/1005.1243

## Rigid and Non-Rigid Mathematical Theories: the Ring

Is Nearly RigidZ(Submitted on 6 May 2010 (v1), last revised 12 May 2010 (this version, v3))Mathematical theories are classified in two distinct classes : {\it rigid}, and on the other hand, {\it non-rigid} ones. Rigid theories, like group theory, topology, category theory, etc., have a basic concept – given for instance by a set of axioms – from which all the other concepts are defined in a unique way. Non-rigid theories, like ring theory, certain general enough pseudo-topologies, etc., have a number of their concepts defined in a more free or relatively independent manner of one another, namely, with {\it compatibility} conditions between them only. As an example, it is shown that the usual ring structure on the integers

is not rigid, however, it is nearly rigid.Z

https://arxiv.org/abs/math/0505336

## What scalars should we use ?

(Submitted on 16 May 2005)There are compelling historical and mathematical reasons why we ended up, among others in Physics, with using the scalars given by the real or the complex numbers. Recently, however, infinitely many easy to construct and use other algebras of scalars have quite naturally emerged in a number of branches of Applied Mathematics. These algebras of scalars can deal with the long disturbing difficulties encountered in Physics, related to such phenomena as “infinities in Physics”, “re-normalization”, the “Feynman path integral”, and so on.

Specifically, as soon as one is dealing with scalars in algebras which – unlike the reals R and complex numbers C – are no longer Archimedean, one can deal with a large variety of “infinite” quantities and do so within the usual rules and with the usual operations of algebra. Here we present typical constructions of these recently emerged algebras of scalars, most of them non-Archimedean.

https://scholar.google.com.au/scholar?cluster=12458896899325487688&hl=en&as_sdt=0,5

@article{bertram2004differential, title={Differential calculus over general base fields and rings}, author={Bertram, Wolfgang and Gl{\"o}ckner, H and Neeb, Karl-Hermann}, journal={Expositiones Mathematicae}, volume={22}, number={3}, pages={213--282}, year={2004}, publisher={Elsevier Science} }

R. Beattie, H.-P. Butzmann (auth.)-Convergence Structures and Applications to Functional Analysis-Springer Netherlands (2002).pdf

This text offers a rigorous introduction into the theory and methods of convergence spaces and gives concrete applications to the problems of functional analysis. While there are a few books dealing with convergence spaces and a great many on functional analysis, there are none with this particular focus.

The book demonstrates the applicability of convergence structures to functional analysis. Highlighted here is the role of continuous convergence, a convergence structure particularly appropriate to function spaces. It is shown to provide an excellent dual structure for both topological groups and topological vector spaces.

Readers will find the text rich in examples. Of interest, as well, are the many filter and ultrafilter proofs which often provide a fresh perspective on a well-known result.

Audience: This text will be of interest to researchers in functional analysis, analysis and topology as well as anyone already working with convergence spaces. It is appropriate for senior undergraduate or graduate level students with some background in analysis and topology.

@book{book:975803,

title = {Convergence Structures and Applications to Functional Analysis},

author = {R. Beattie, H.-P. Butzmann (auth.)},

publisher = {Springer Netherlands},

isbn = {978-90-481-5994-9,978-94-015-9942-9},

year = {2002},

series = {},

edition = {1},

volume = {},

url = {http://gen.lib.rus.ec/book/index.php?md5=9c36167c938098065b764ddf32889ddd}

}

(Lecture Notes in Mathematics 469) Prof. Dr. Ernst Binz (auth.)-Continuous Convergence on C(X)-Springer-Verlag Berlin Heidelberg (1975).pdf

@book{book:973250,

title = {Continuous Convergence on C(X)},

author = {Prof. Dr. Ernst Binz (auth.)},

publisher = {Springer-Verlag Berlin Heidelberg},

isbn = {978-3-540-07179-2,978-3-540-37519-7},

year = {1975},

series = {Lecture Notes in Mathematics 469},

edition = {1},

volume = {},

url = {http://gen.lib.rus.ec/book/index.php?md5=083542c932efb1ee4000123987ce06e6}

}

## Approach Theory

R. Lowen-Index Analysis_ Approach Theory at Work-Springer (2015).pdf

@book{book:2201083,

title = {Index Analysis: Approach Theory at Work},

author = {R. Lowen},

publisher = {Springer},

isbn = {1447164849,9781447164845},

year = {2015},

series = {},

edition = {2015},

volume = {},

url = {http://gen.lib.rus.ec/book/index.php?md5=82e4edc658e1e92d8bbc60e83d52ddb0}

}

(Oxford Mathematical Monographs) R. Lowen-Approach Spaces_ The Missing Link in the Topology-Uniformity-Metric Triad-Oxford University Press, USA (1997).pdf

@book{book:186303,

title = {Approach Spaces: The Missing Link in the Topology-Uniformity-Metric Triad},

author = {R. Lowen},

publisher = {Oxford University Press, USA},

isbn = {0198500300,9780198500308},

year = {1997},

series = {Oxford Mathematical Monographs},

edition = {},

volume = {},

url = {http://gen.lib.rus.ec/book/index.php?md5=5946C5017DD8A72F0868A7C4D31BA0FB}

}