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.
