**Algebraic General Topology. Vol 1**:
Paperback
/
E-book
||
**Axiomatic Theory of Formulas**:
Paperback
/
E-book

This document contains a list of short ideas of future research in Algebraic

General Topology.

I have created branch devel in the L

A

T

E

Xrepository for the book to add new

“draft” features there. The devel branch isn’t distributed by me in PDF format,

but you can download and compile it yourself.

This research plan is not formal and may contain vague statements.

1. Things to do first

Isn’t generalized limit just the limit on the set of “singularities”? If yes, it seems

a key to put it into a diﬀeq!

Which ﬁlter operations are congruences on equivalence of ﬁlters?

2. Misc

Both uniform covers and functions can be represented as sets of binary Cartesian

products (uniform covers as sets of “quadrtatic” products, function as sets of prod-

ucts of singletons). Deﬁne composition as ??. Therefore we can form a semigroup

of them. What is the action of this semigroup?

Some special cases of reloids: https://www.researchgate.net/publication/

331776637 Functional Boundedness of Balleans Coarse Versions of Compactness

“Unﬁxed” for more general settings than lattice and its sublattice. (However, it

looks like this generalization has no practical applications.)

Should clearly denote pFCD(A; Z) or pFCD(A).

https://en.wikipedia.org/wiki/Compact element

https://arxiv.org/abs/1904.12525 On proximal ﬁneness of topological groups in

their right uniformity

https://arxiv.org/abs/1905.00513 On B-Open Sets

https://arxiv.org/abs/1812.09802 Boundaries of coarse proximity spaces and

boundaries of compactiﬁcations

Try to describe a ﬁlter with up of inﬁntiely small components. For this use a

ﬁlter (of sets or ﬁlters) rather than a set of sets.

About generalization of simplical sets for nearness spaces on posets? https:

//arxiv.org/abs/1902.07948

3. Category theory

Can product morphism (in a category with restricted identities) be considered as

a categorical product in arrow category? (It seems impossible to deﬁne projections

for arbitrary categories with binary product morphism. Can it be in the special

cases of funcoids and reloids?)

Attempting to extend Tychonoﬀ product from topologies to funcoids: —— If i

has left adjoint: If r is left adjoint to i, we have Hom(A, i(X ×Y )) = Hom(r(A), X ×

Y ) = Hom(r(A), X) × Hom(r(A), Y ) = Hom(A, i(X)) × Hom(A, i(Y )). —— If also

the left adjoint is full and faithful: Hom(A, i(r(X) × r(Y ))) = Hom(r(A), r(X) ×

r(Y )) = Hom(r(A), r(X)) × Hom(r(A), r(Y )) = Hom(A, X) × Hom(A, Y ). See

also http://math.stackexchange.com/q/1982931/4876. However this does not apply

because reﬂection of topologies in funcoids is not full.

Being intersecting is deﬁned for posets (= thin categories). It seems that this

can be generalized for any categories. This way we can deﬁne (pointfree) funcoids

1

2

between categories generalizing pointfree funcoids between posets. (However this

is probably easily reducible to the case of posets.)

I have deﬁned RLD] to describe Hom-sets of the category or reloids but without

source and destination and without composition. RLD should be replaced with

RLD] where possible, in order to make the theorems throughout the book a little

more general. Also introduce similar features like Γ] and FΓ] (the last notation

may need to be changed).

Misc properties of continuous functions between endofuncoids and endoreloids.

http://nforum.ncatlab.org/discussion/6765/please-help-with-a-proof-that-a-category-is-monoidal/

proves that ﬁnitary staroids are isomorphic to an ideal on a poset (for semilattices

only).

[1] deﬁnes two categories with objects being ﬁlters. Another article on the same

topic:

https://eudml.org/doc/16352 (Koubek, V´aclav, and Reiterman, Jan. ”On the cat-

egory of ﬁlters.”)

FiXme: https://en.wikipedia.org/wiki/Cauchyspace says “The category of

Cauchy spaces and Cauchy continuous maps is cartesian closed.” Generalize.

http://www.sciencedirect.com/science/article/pii/0166864187900988

4. Compact funcoids

Generalize the theorem that compact topology corresponds to only one unifor-

mity.

For compact funcoids the Cantor’s theorem that a function continuous on a

compact is uniformly continuous.

Every closed subset of a compact space is compact. A compact subset of a

Hausdorﬀ space is closed. 17.5 theorem in Willard.

17.6 theorem in Willard.

17.7 theorem in Willard: The continuous image of a compact space is compact.

17.10 Theorem in Willard: A compact Hausdorﬀ space X is a T 4-space. Also

17.11 Corollary, 17.13, 17.14 theorem.

”Locally compact” for funcoids. See also 18 ”Locally compact spaces: in Willard.

Compactiﬁcation.

5. Misc

A funcoid or pointfree funcoids can be turned into a semigroup action also by

the formula: hfi(x, y) = (hfi(x),

f

−1

(y)) (on the left hi denotes the semigroup

action, on the right it denotes components of the funcoid.)

Counterexample at https://math.stackexchange.com/a/3046071/4876.

https://www.researchgate.net/project/Contra-continuity-in-its-diﬀerent-aspects

We know that (RLD)

out

(f t g) = (RLD)

out

f t (RLD)

out

g. Hm, then it is a

pointfree funcoid!

Conjecture 1. hfi

d

S =

d

X ∈S

hfiX if S is a totally ordered (generalize for a ﬁlter

base) set of ﬁlters (or at least set of sets). [Counterexample: https://portonmath.

wordpress.com/2018/05/20/a-counterexample-to-my-recent-conjecture/]

Should we replace the word “intersect” with the word “overlap”?

Instead of a ﬁltrator use “closure“ (X, [X])?

3

(FCD), (RLD)

in

, (RLD)

out

can be deﬁned purely in terms of ﬁltrators. So gener-

alize it.

Generalize for funcoids and reloids factoring into monovalued and injective:

https://math.stackexchange.com/q/2414159/4876. Generalize it for

star-composition with multidimensional, identity relations, identity

staroids/multifuncoids, or identity reloid. Isn’t thus a category with star-

morphisms determined by a regular category?! Also try to split into complete and

co-complete funcoids/reloids.

Open problems on βω (Klass Pieter Hart and Jab van Mill).

Example that Compl f t CoCompl f @ f (for both funcoids and reloids). Proof

for funcoids (for reloids it’s similar): Take f = A ×

FCD

B. Then (write an ex-

plicit proof) Compl f = (Cor A) ×

FCD

B and CoCompl f = A ×

FCD

(Cor B). Thus

Compl f t CoCompl f 6= f (if A, B are non-principal).

Every funcoid (reloid) is a join of monovalued funcoids (reloids). For funcoids

it’s obvious (because it’s a join of atomic funcoids). For reloids?

“Vicinity” and “neighborhood” mean diﬀerent things, e.g. in [2].

Micronization µ and S

∗

are in some sense related as Galois connection. To

formalize this we need to extend µ to arbitrary reloids (not only binary relations).

We need (it is especially important for studying compactness) to ﬁnd a product

of funcoids which coincides with product of topological spaces. (Cross-composition

product doesn’t because it is even not a funcoid (but a pointfree funcoid).) Neither

subatomic product.

Subspace topology for space µ and set X is equal to µ u (X ×

FCD

X).

Change terminology: monotone → increasing.

What are necessary and suﬃcient conditions for up f to be a ﬁlter for a funcoid f?

Article “Neighborhood Spaces” by D. C. KENT and WON KEUN MIN. ftp:

//ftp.math.ethz.ch/EMIS/journals/IJMMS/Volume327/239107.pdf

g v f

◦

◦ h ⇔ f ◦ g v h?

lim x → af(x) = b iﬀ x → a implies hf ix → b for all ﬁlters x.

http://mathoverﬂow.net/q/36999/4086 “A good place to read about uniform

spaces”.

Research the posets of all proximity spaces and all uniform spaces (and also

possibly reﬂexive and transitive funcoids/reloids).

Are ﬁlters on all Heyting or all co-Heyting lattices star-separable? http://math.

stackexchange.com/q/1326266/4876

Deﬁne generalized pointfree reloids as ﬁlters on systems of sides.

Galois connections primer – study to ensure that we considered all Galois con-

nections properties.

Germs seems to be equivalent to monovalued reloids.

A = min

n

X

∀K∈∂A:K6K

o

, so we can restore A from ∂A.

Boolean funcoid is a join-semilattice morphism from a boolean lattice to a

boolean lattice. Generalize for pointfree funcoids.

Another way to deﬁne pointfree reloid as ﬁlters on Galois connections between

two posets.

L ∈ GR

Q

Strd∗

A ⇔ ∀ﬁnite M ⊆ dom A∀i ∈ M : Ai 6 Li?

Star-composition with identity staroids?

Does upgrading/downgrading of the ideal which represents a prestaroid coincide

with upgrading/downgrading of the prestaroid?

4

It seems that equivalence of ﬁlters on diﬀerent bases can be generalized: ﬁl-

ters A ∈ A and B ∈ B are equivalent iﬀ there exists an X ∈ A ∩ B which is greater

than both A and B. This however works only in the case if order of the orders A

and B agree, that is if then are both a suborders of a greater ﬁxed order.

Under which conditions a function spaces of posets is strongly separable?

Generalize both funcoids and reloids as ﬁlters on a superset of the lattice Γ (see

“Funcoids are ﬁlters” chapter).

When the set of ﬁlters closed regarding a funcoid is a (co-)frame?

If a formula F (x0, . . . , xn) holds for every poset Ai then it also holds for product

order

Q

A. (What about inﬁnite formulas like complete lattice joins and meets?)

Moreover F (x0, . . . , xn) = λi ∈ n : F (x0, i, . . . , xn, i) (confused logical forms and

functions). It looks like a promising approach, but how to deﬁne it exactly? For

example, F may be a form always true for boolean lattices or for Heyting lattices,

or whatsoever. How one theorem can encompass all kinds of lattices and posets?

We may attempt to restrict to (partial) functions determined by order. (This is not

enough, because we can deﬁne an operation restricting \ deﬁned only for posets

of cardinality above or below some cardinal κ. For such restricted \ the above

formula does not work.) See also https://portonmath.wordpress.com/2016/01/

12/a-conjecture-about-product-order-and-logic/. It seems that Todd Trimble

shows a general category-theoretic way to describe this: https://nforum.ncatlab.

org/discussion/6887/operations-on-product-order/.

Get results from http://ncatlab.org/toddtrimble/published/topogeny.

What about distributivity of quasicomplements over meets and joins for the

ﬁltrator of funcoids? Seems like nontrivial conjectures.

Conjecture: Each ﬁltered ﬁltrator is isomorphic to a primary ﬁltrator. (If it

holds, then primary and ﬁltered ﬁltrators are the same!)

Add analog of the last item of the theorem about co-complete funcoids for point-

free funcoids.

Generalize theorems about RLD(A; B) as F (A × B) in order to clean up the

notation (for example in the chapter “Funcoids are ﬁlters”).

Deﬁne reloids as a ﬁltrator whose core is an ordered semigroup. This way reloids

can be described in several isomorphic ways (just like primary ﬁltrators are both

ﬁltrators of ﬁlters, of ideals, etc.) Is it enough to describe all properties of reloids?

Well, it is not a semigroup, it is a precategory. It seems that we also need functions

dom and im into partially ordered sets and “reversion” (dagger).

http://mathoverﬂow.net/a/191381/4086 says that n-staroids can be identiﬁed

with certain ideals!

To relax theorem conditions and deﬁnition, we can deﬁne protofuncoids as arbi-

trary pairs (α; β) of functions between two posets. For protofuncoids composition

and reverse are deﬁned.

Add examples of funcoids to demonstrate their power: D t T (D is a digraph T

is a topological space), T u

n

(x;y)

y≥x

o

as “one-side topology” and also a circle made

from its π-length segment.

Say explicitly that pseudodiﬀerence is a special case of diﬀerence.

For pointfree funcoids, if f : A → B exists, then existence of least element of A

is equivalent to existence of least element of B: y 6 hfi⊥

A

⇔ ⊥

A

6 hf

−1

iy ⇔ 0.

Thus hf iy hfiy and so hf iy = ⊥

B

. Can a similar statement be made that A being

join-semilattice implies B being join-semilattice (at least for separable posets)? If

5

yes, this could allow to shorten some theorem conditions. It seems we can produce

a counter-example for non-separable posets by replacing an element with another

element with the same full star.

Develop Todd Trimble’s idea to represent funcoids as a relation ξ further: Deﬁne

funcoid as a function from sets to sets of sets ξ(A t B) = ξA ∩ ξB and ξ⊥ = ∅.

Denote the set of least elements as Least. (It is either an one-element set or

empty set.)

Show that cross-composition product is a special case of inﬁmum product.

Analog of order topology for funcoids/reloids.

A set is connected if every function from it to a discrete space is constant. Can

this be generalized for generalized connectedness and generalized continuity? I have

no idea how to relate these two concepts in general.

Develop theory of funcoidal groups by analogy with topological groups. Attempt

to use this theory to solve this open problem:

http://garden.irmacs.sfu.ca/?q=op/iseveryregularparatopologicalgrouptychonoﬀ

Is it useful as topological group determines not only a topology but

even a uniformity? An interesting article on topological groups: https:

//arxiv.org/abs/1901.01420

Consider generalizations of this article:

https://www.researchgate.net/publication/318822240 Categorically Closed

Topological Groups

A space µ is T 2- iﬀ the diagonal ∆ is closed in µ × µ.

The β-th projection map is not only continuous but also open (Willard, theorem

8.6).

T x-separation axioms for products of spaces.

Willard 13.13 and its important corollary 13.14.

Willard 15.10.

About real-valued functions on endofuncoids: Urysohn’s Lemma (and conse-

quences: Tietze’s extension theorem) for funcoids.

About product of reloids:

http://portonmath.wordpress.com/2012/05/23/unﬁrunded-questions/

Generalized Fr´echet ﬁlter on a poset (generalize for ﬁltrators) A is a ﬁlter Ω such

that

∂Ω =

x ∈ A

atoms x is inﬁnite

.

Research their properties (ﬁrst, whether they exists for every poset). Also consider

Fr´echet element of FCD(A; B). Another generalization of Fr´echet ﬁlter is meet of

all coatoms.

Manifolds.

http://www.sciencedirect.com/science/article/pii/0304397585900623

(free download, also Google for ”pre-adjunction”, also ”semi” instead of ”pre”) Are

(FCD) and (RLD)

in

adjunct?

Check how multicategories are related with categories with star-morphisms.

At https://en.wikipedia.org/wiki/Semilattice they are deﬁned distributive semi-

lattices. A join-semilattice is distributive if and only if the lattice of its ideals

(under inclusion) is distributive.

6

The article http://arxiv.org/abs/1410.1504 has solved “Every paratopological

group is Tychonoﬀ” conjecture positively. Rewrite this article in terms of funcoids

and reloids (especially with the algebraic formulas characterizing regular funcoids).

Generalize interior in topological spaces as the interior funcoid of a co-complete

funcoid f, deﬁned as a pointfree funcoid f

◦

: F dual Src f → F dual Dst f conform-

ing to the formula: hf

◦

i

∗

(I u J) = hf i

∗

I u J = hfi

∗

(I t J ). However composition

of an interior funcoid with a funcoid is neither a funcoid nor an interior funcoid. It

can be generalized using pseudocomplement.

http://math.sun.ac.za/cattop/Output/Kunzi/quasiintr.pdf “An Introduction to

the Theory of Quasi-uniform Spaces”.

http://www.mscand.dk/article/download/10581/8602 (“On equivalence be-

tween proximity structures and totally bounded uniform structures”)

Characterize the set

n

f∈FCD

(RLD)

in

f=(RLD)

out

f

o

. (This seems a diﬃcult problem.) An-

other (possibly related) problem: when up f is a ﬁlter for a funcoid f .

Deﬁne S

∗

(f) for a funcoid f (using that f is a ﬁlter).

Let A be a ﬁlter. Is the boolean algebra Z(DA) a. atomic; b. complete?

https://arxiv.org/pdf/1003.5377.pdf

https://www.researchgate.net/project/Generalized-topological-groups-in-Delfs-Knebusch-generalized-topology

and http://www.sciencedirect.com/science/authShare/S0166864117302742/

20170530T163200Z/1?md5=a5f9bcce5a6c49d4b8b35fdc0d2f9105 (not available for

free).

https://arxiv.org/abs/1802.05746 about uniform spaces and function spaces.

https://arxiv.org/abs/1904.08969 about k-Scattered spaces.

https://www.researchgate.net/publication/333731858 SUPERCOMPACT

MINUS COMPACT IS SUPER seems interesting.

“Second reloidal product” of more than two ﬁlters. Also starred second product.

Homotopy with a monovalued (complete?) funcoid from R instead of path.

What’s about limits multidimesional functions? ∀x

i

: x

i

→ α

i

⇒ f(x) → β.

“Contra continuity” (see journals).

Product of pointfree funcoids considered as structures: http://citeseerx.ist.psu.

edu/viewdoc/download?doi=10.1.1.414.9364&rep=rep1&type=pdf. 2-staroids are

universal classes: https://modeltheory.fandom.com/wiki/Universal theory

http://imar.ro/journals/Revue Mathematique/pdfs/2015/2/5.pdf “ON

IDEALS AND FILTERS IN POSETS” SERGIU RUDEANU.

https://www.researchgate.net/publication/334695188 A note on compact-like

semitopological groups A note on compact-like semitopological groups.

https://arxiv.org/abs/1907.12129 Closed subsets of compact-like topological

spaces.

https://arxiv.org/abs/1908.05624 A remark on locally direct product subsets in

a topological Cartesian space.

https://arxiv.org/abs/1909.06428 Coproducts of proximity spaces.

https://arxiv.org/abs/1909.09303 On T

0

spaces determined by well-ﬁltered

spaces.

https://arxiv.org/abs/1812.06379 Closed Discrete Selection in the Compact

Open Topology.

https://www.researchgate.net/publication/333731858 SUPERCOMPACT

MINUS COMPACT IS SUPER Supercompact minus compact is super.

7

https://www.jstor.org/stable/2306387?read-now=1&seq=1 Ideals in partially

ordered sets (free to read).

https://blog.mathematics21.org/2019/11/07/funcoid-is-a-structure-in-the-sense-of-math-logic/

(blog post).

https://arxiv.org/abs/1910.01014 Codensity: Isbell duality, pro-objects, com-

pactness and accessibility.

https://arxiv.org/abs/1910.12228 New proofs for some fundamental results of

topology.

https://arxiv.org/abs/1910.12228 A simple proof of Tychonoﬀ theorem.

https://arxiv.org/abs/1910.05293 Lusin and Suslin properties of function spaces.

https://www.researchgate.net/project/Generalized-topological-groups-in-Delfs-Knebusch-generalized-topology

Generalized topological groups in Delfs-Knebusch generalized topology.

https://www.researchgate.net/publication/334759733

Closed subsets of

compact-like topological spaces Closed subsets of compact-like topological

spaces.

ttps://arxiv.org/abs/1906.10832 Existence of well-ﬁlteriﬁcations of T

0

topologi-

cal spaces.

https://arxiv.org/abs/1906.11194 Locally ordered topological spaces.

https://arxiv.org/abs/1906.03549 Supercompact minus compact is super.

https://arxiv.org/abs/1905.12446 On H

Y

-Ideals.

https://arxiv.org/abs/1910.12228 New proofs for some fundamental results of

topology.

https://arxiv.org/abs/1906.08498 A T

0

Compactiﬁcation Of A Tychonoﬀ Space

Using The Rings Of Baire One Functions.

https://arxiv.org/abs/1911.05390 Soft T

(0,α)

Spaces.

6. Common generalizations of funcoids and convergence spaces

I propose the following (possible) common generalizations of funcoids and con-

vergence spaces ([2]):

• To every set we associate an isotone (and in some sense preserving ﬁnite

joins) collection of ﬁlters.

• To every ﬁlter we associate an isotone (and in some sense preserving ﬁnite

joins) collection of ﬁlters.

• Consider pointfree funcoids between isotone families of ﬁlters.

• What’s about “double-ﬁltrator” (A; B; C)?

7. Dimension

Deﬁne dimensions of funcoids.

Every funcoid of dimension N can be represented as a subfuncoid of a composi-

tion f

n

◦ · · · ◦ f

0

of N -planes f

0

, . . . , f

n

? Seems wrong, counterexample:

S

i∈N

l

1

i+1

where l

α

is the abscissa rotated α radians.

Don’t forget to add open problems to http://openproblemgarden.org and https:

//scilag.net.

References

[1] Andreas Blass. Two closed categories of ﬁlters. Fundamenta Mathematicae, 94(2):129–143,

1977.

8

[2] Szymon Dolecki and Fr´ed´eric Mynard. Convergence Foundations of Topology. World Scientiﬁc,

2016.