Variable-basis topological systems Sergejs Solovjovs University of - - PowerPoint PPT Presentation
Motivation Preliminaries Topological systems Systems & Spaces Problems Variable-basis topological systems Sergejs Solovjovs University of Latvia International Category Theory Conference 2008 Calais, France June 22 - 28, 2008
Motivation Preliminaries Topological systems Systems & Spaces Problems
Sergejs Solovjovs
University of Latvia
International Category Theory Conference 2008 Calais, France June 22 - 28, 2008
Variable-basis topological systems Sergejs Solovjovs University of Latvia 1/42
Motivation Preliminaries Topological systems Systems & Spaces Problems
1
Motivation
2
Algebraic and topological preliminaries
3
Variable-basis topological systems
4
Topological systems versus topological spaces
5
Open problems
Variable-basis topological systems Sergejs Solovjovs University of Latvia 2/42
Motivation Preliminaries Topological systems Systems & Spaces Problems Historical remarks
1959 D. Papert and S. Papert construct an adjunction between the categories Top (topological spaces) and Frmop (the dual of the category Frm of frames). 1972 J. Isbell uses the name locale for the objects of Frmop and considers the category Loc (locales) as a substitute for Top. 1982 P. Johnstone gives a coherent statement to localic theory in his book “Stone Spaces”. 1989 Using the logic of finite observations S. Vickers introduces the notion of topological system to unite both topological and localic approaches.
Variable-basis topological systems Sergejs Solovjovs University of Latvia 3/42
Motivation Preliminaries Topological systems Systems & Spaces Problems Historical remarks
1965 L. A. Zadeh introduces fuzzy sets. His approach is generalized by J. A. Goguen in 1967. 1968 C. L. Chang introduces fuzzy topological spaces. His approach is generalized by R. Lowen in 1976. 1983 S. E. Rodabaugh studies the category FUZZ of variable-basis fuzzy topological spaces. Later on he considers the category C-Top of variable-basis lattice-valued topological spaces. . . . Starting from 1983 U. H¨
Sostak et al. consider fixed- and variable-basis fuzzy topologies and their properties.
Variable-basis topological systems Sergejs Solovjovs University of Latvia 4/42
Motivation Preliminaries Topological systems Systems & Spaces Problems Historical remarks
2007 J. T. Denniston and S. E. Rodabaugh consider functorial relationships between lattice-valued topology and topological systems. !!! Using fuzzy topological spaces and crisp topological systems they encounter some problems.
Variable-basis topological systems Sergejs Solovjovs University of Latvia 5/42
Motivation Preliminaries Topological systems Systems & Spaces Problems
2008 We introduced the category of variable-basis topological spaces over an arbitrary variety of algebras generalizing the category C-Top of S. E. Rodabaugh. This talk introduces the notion of variable-basis topological system over an arbitrary variety of algebras. By analogy with J. T. Denniston and S. E. Rodabaugh we consider functorial relationships between variable-basis topological spaces and variable-basis topological systems. The basic point While considering fuzzy topological spaces, one should consider fuzzy topological systems.
Variable-basis topological systems Sergejs Solovjovs University of Latvia 6/42
Motivation Preliminaries Topological systems Systems & Spaces Problems Varieties of algebras
Let Ω = (nλ)λ∈Λ be a class of cardinal numbers. Definition 1 An Ω-algebra is a pair (A, (ωA
λ)λ∈Λ) (denoted by A), where A
is a set and (ωA
λ)λ∈Λ is a family of maps Anλ ωA
λ
− → A. An Ω-homomorphism (A, (ωA
λ)λ∈Λ) f
− → (B, (ωB
λ )λ∈Λ) is a map
A f − → B such that f ◦ ωA
λ = ωB λ ◦ f nλ for every λ ∈ Λ.
Definition 2 Alg(Ω) is the category of Ω-algebras and Ω-homomorphisms. | − | is the forgetful functor to the category Set (sets).
Variable-basis topological systems Sergejs Solovjovs University of Latvia 7/42
Motivation Preliminaries Topological systems Systems & Spaces Problems Varieties of algebras
Definition 3 Let M (resp. E) be the class of Ω-homomorphisms with injective (resp. surjective) underlying maps. A variety of Ω-algebras is a full subcategory of Alg(Ω) closed under the formation of products, M-subobjects (subalgebras) and E-quotients (homomorphic images). The objects (resp. morphisms) of a variety are called algebras (resp. homomorphisms). Example 4 The categories Frm, SFrm and SQuant of frames, semiframes and semi-quantales (popular in lattice-valued topology) are varieties.
Variable-basis topological systems Sergejs Solovjovs University of Latvia 8/42
Motivation Preliminaries Topological systems Systems & Spaces Problems Fixed-basis topology
From now one fix a variety A and an algebra Q. Definition 5 Given a set X, QX is the Q-powerset of X. An arbitrary element of QX is denoted by p (with indices). QX is an algebra with operations lifted point-wise from Q by (ωQX
λ (pinλ))(x) = ωQ λ (pi(x)nλ).
Variable-basis topological systems Sergejs Solovjovs University of Latvia 9/42
Motivation Preliminaries Topological systems Systems & Spaces Problems Fixed-basis topology
Let X
f
− → Y be a map and let A
g
− → B be a homomorphism. There exist:
the standard image and preimage operators P(X)
f →
− − → P(Y ) and P(Y )
f ←
− − → P(X); the Zadeh preimage operator QY
f ←
Q
− − → QX defined by f ←
Q (p) = p ◦ f ;
a map AX
g X
→
− − → BX defined by g X
→(p) = g ◦ p.
Lemma 6 For every map X
f
− → Y and every homomorphism A
g
− → B, both QY
f ←
Q
− − → QX and AX
gX
→
− − → BX are homomorphisms.
Variable-basis topological systems Sergejs Solovjovs University of Latvia 10/42
Motivation Preliminaries Topological systems Systems & Spaces Problems Fixed-basis topology
Definition 7 Given a set X, a subset τ of QX is a Q-topology on X provided that τ is a subalgebra of QX. A Q-topological space (also called a Q-space) is a pair (X, τ), where X is a set and τ is a Q-topology on X. A map (X, τ) f − → (Y , σ) between Q-spaces is Q-continuous provided that (f ←
Q )→(σ) ⊆ τ.
Definition 8 Q-Top is the category of Q-spaces and Q-continuous maps. | − | is the forgetful functor to the category Set.
Variable-basis topological systems Sergejs Solovjovs University of Latvia 11/42
Motivation Preliminaries Topological systems Systems & Spaces Problems Variable-basis topology
From now on introduce the following notations: The dual of the category A is denoted by LoA (the “Lo” comes from “localic”). The objects (resp. morphisms) of LoA are called localic algebras (resp. homomorphisms). The respective homomorphism of a localic homomorphism f is denoted by f op and vice versa. To distinguish between maps and homomorphisms denote them by “f , g” and “ϕ, ψ” respectively.
Variable-basis topological systems Sergejs Solovjovs University of Latvia 12/42
Motivation Preliminaries Topological systems Systems & Spaces Problems Variable-basis topology
Definition 9 Given a Set × LoA-morphism (X, A)
(f ,ϕ)
− − − → (Y , B), there exists the Rodabaugh preimage operator BY
(f ,ϕ)←
− − − − → AX defined by (f , ϕ)←(p) = ϕop ◦ p ◦ f . Lemma 10 For every Set × LoA-morphism (X, A)
(f ,ϕ)
− − − → (Y , B), the diagram BY
f ←
B
→
f ←
A
(ϕop)X
→
AX
commutes and therefore BY
(f ,ϕ)←
− − − − → AX is a homomorphism.
Variable-basis topological systems Sergejs Solovjovs University of Latvia 13/42
Motivation Preliminaries Topological systems Systems & Spaces Problems Variable-basis topology
Definition 11 Given a subcategory C of LoA, the category C-Top comprises the following data:
Objects: C-topological spaces or C-spaces (X, A, τ), where (X, A) is a Set × C-object and (X, τ) is an A-space. Morphisms: C-continuous pairs (X, A, τ)
(f ,ϕ)
− − − → (Y , B, σ), where (f , ϕ) is a Set × C-morphism and ((f , ϕ)←)→(σ) ⊆ τ.
| − | is the forgetful functor to the category Set × C. C-Top generalizes the respective category of S. E. Rodabaugh. This talk considers the case C = LoA. Call LoA-spaces by spaces and LoA-continuity by continuity.
Variable-basis topological systems Sergejs Solovjovs University of Latvia 14/42
Motivation Preliminaries Topological systems Systems & Spaces Problems Topological systems of S. Vickers
Definition 12 Let X be a set and A be a frame. Then X
| =
− → A is a satisfaction relation on (X, A) if | = is a binary relation from X to A satisfying the following join interchange law and meet interchange law: For any family {ai}i∈I of elements of A, x | =
i∈I
ai iff x | = ai for at least one i ∈ I. For any finite family {ai}i∈I of elements of A, x | =
i∈I
ai iff x | = ai for every i ∈ I. If x | = a, then x satisfies a.
Variable-basis topological systems Sergejs Solovjovs University of Latvia 15/42
Motivation Preliminaries Topological systems Systems & Spaces Problems Topological systems of S. Vickers
Definition 13 A topological system is a triple (X, A, | =), where (X, A) is a Set × Loc-object and | = is a satisfaction relation on (X, A). Elements of X are points and elements of A are opens. The category TopSys comprises the following data:
Objects: topological systems (X, A, | =). Morphisms: continuous maps (X, A, | =1)
f =(pt f ,(Ωf )op)
− − − − − − − − − → (Y , B, | =2), where f is a Set × Loc-morphism and for every x ∈ X, b ∈ B, pt f (x) | =2 b iff x | =1 Ωf (b).
Variable-basis topological systems Sergejs Solovjovs University of Latvia 16/42
Motivation Preliminaries Topological systems Systems & Spaces Problems Variable-basis approach
Definition 14 Given a subcategory C of LoA, the category C-TopSys comprises the following data:
Objects: C-topological systems or C-systems (X, A, B, | =), where (X, A, B) is a Set × C × C-object and X × B
| =
− → A is a map (satisfaction relation) such that for every x ∈ X, B
| =(x,−)
− − − − → A is a homomorphism. Morphisms: C-continuous maps (X, A, B, | =1)
f =(pt f ,(Σf )op,(Ωf )op)
− − − − − − − − − − − − − → (Y , C, D, | =2), where f is a Set × C × C-morphism and for every x ∈X, d ∈D, Σf (| =2(pt f (x), d)) = | =1(x, Ωf (d)).
| − | is the forgetful functor to the category Set × C × C.
Variable-basis topological systems Sergejs Solovjovs University of Latvia 17/42
Motivation Preliminaries Topological systems Systems & Spaces Problems Variable-basis approach
Definition 15 For a C-object Q, Q-TopSys is the subcategory of C-TopSys
=) with basis Q and all continuous f such that Σf = 1Q. | − | is the forgetful functor to the category Set × C. Lemma 16 The subcategory Q-TopSys is full iff C(Q, Q) = {1Q}. If Q is an initial (terminal) object in A, then Q-TopSys is full.
Variable-basis topological systems Sergejs Solovjovs University of Latvia 18/42
Motivation Preliminaries Topological systems Systems & Spaces Problems Variable-basis approach
Example 17 2 = {⊥, ⊤} is initial in Frm. The full subcategory 2-TopSys of Loc-TopSys is isomorphic to the category TopSys of S. Vickers. Example 18 Given a set K, the subcategory K-TopSys of LoSet-TopSys is isomorphic to the category Chu(Set, K) of Chu spaces over K. K-TopSys is full iff K is the empty set or a singleton. The following considers the category LoA-TopSys. Call LoA-systems by systems and LoA-continuity by continuity.
Variable-basis topological systems Sergejs Solovjovs University of Latvia 19/42
Motivation Preliminaries Topological systems Systems & Spaces Problems Topological spaces versus topological systems
Lemma 19 There exists a full embedding LoA-Top
ET LoA-TopSys with
ET((X, A, τ)
(f ,ϕ)
− − − → (Y , B, σ)) = (X, A, τ, | =1)
(f ,ϕ,((f ,ϕ)←)op)
− − − − − − − − − − → (Y , B, σ, | =2) where | =i(z, p) = p(z). Proof. As an example show that ET(f , ϕ) is in LoA-TopSys: | =1(x, (f , ϕ)←(p)) = | =1(x, ϕop ◦ p ◦ f ) = ϕop ◦ p ◦ f (x) = ϕop(| =2(f (x), p)).
Variable-basis topological systems Sergejs Solovjovs University of Latvia 20/42
Motivation Preliminaries Topological systems Systems & Spaces Problems Topological spaces versus topological systems
Lemma 20 There exists a functor LoA-TopSys
Spat
− − → LoA-Top defined by Spat((X, A, B, | =1) f − → (Y , C, D, | =2)) = (X, A, τ)
(pt f ,(Σf )op)
− − − − − − − → (Y , C, σ) where τ = {| =1(−, b) | b ∈ B} (| =1(−, b) is the extent of b). Proof. As an example show that Spat(f ) is in LoA-Top: ((pt f , (Σf )op)←(| =2(−, d)))(x) = Σf ◦ | =2(−, d) ◦ pt f (x) = Σf (| =2(pt f (x), d)) = | =1(x, Ωf (d)) = (| =1(−, Ωf (d)))(x).
Variable-basis topological systems Sergejs Solovjovs University of Latvia 21/42
Motivation Preliminaries Topological systems Systems & Spaces Problems Topological spaces versus topological systems
Theorem 21 Spat is a right-adjoint-left-inverse of ET. Proof. Given a system (X, A, B, | =), ET Spat(X, A, B, | =)
(1X ,1A,Φop)
− − − − − − − → (X, A, B, | =) with Φ(b) = | =(−, b) provides an ET-(co-universal) map. Straightforward computations show that Spat ET = 1LoA-Top. Corollary 22 LoA-Top is isomorphic to a full (regular mono)-coreflective subcategory of LoA-TopSys.
Variable-basis topological systems Sergejs Solovjovs University of Latvia 22/42
Motivation Preliminaries Topological systems Systems & Spaces Problems Localic algebras versus topological systems
Lemma 23 There exists an embedding LoA
E Q
L
− − → LoA-TopSys defined by E Q
L (B ϕ
− → C) = (PtQ(B), Q, B, | =1)
((ϕop)←
Q ,1Q,ϕ)
− − − − − − − − − → (PtQ(C), Q, C, | =2) where PtQ(B) = A(B, Q) and | =i(p, d) = p(d). E Q
L is full iff A(Q, Q) = {1Q}.
If Q is an initial (terminal) object in A, then E Q
L is full.
Variable-basis topological systems Sergejs Solovjovs University of Latvia 23/42
Motivation Preliminaries Topological systems Systems & Spaces Problems Localic algebras versus topological systems
Lemma 24 There exists a functor LoA-TopSys Loc − − → LoA defined by Loc((X, A, B, | =1) f − → (Y , C, D, | =2)) = B
(Ωf )op
− − − − → D. Lemma 25 In general E Q
L does not have a left adjoint and therefore Loc is not
a left adjoint of E Q
L .
Variable-basis topological systems Sergejs Solovjovs University of Latvia 24/42
Motivation Preliminaries Topological systems Systems & Spaces Problems Other functorial relationships
Lemma 26 Every homomorphism A
ϕ
− → B provides an embedding A-Top
Eϕ LoA-TopSys defined by
Eϕ((X, τ) f − → (Y , σ)) = (X, B, τ, | =ϕ
1 ) (f ,1B,(f ←
A )op)
− − − − − − − − → (Y , B, σ, | =ϕ
2 )
where | =ϕ
i (z, p) = ϕ ◦ p(z).
If ϕ is an A-monomorphism and A(B, B) = {1B}, Eϕ is full.
Variable-basis topological systems Sergejs Solovjovs University of Latvia 25/42
Motivation Preliminaries Topological systems Systems & Spaces Problems Other functorial relationships
Lemma 27 There exists a functor LoA-TopSys
F|
=
− − → LoA-Top defined by F|
=((X, A, B, |
=1) f − → (Y , C, D, | =2)) = (X, B, τ)
(pt f ,(Ωf )op)
− − − − − − − → (Y , D, σ) where τ = {p ∈ BX | | =1(x, p(x)) = | =1(x′, p(x′)) for every x, x′ ∈ X}. F|
= is related to stratified topological spaces of R. Lowen.
In contrast to Spat, F|
= forgets the basis.
Variable-basis topological systems Sergejs Solovjovs University of Latvia 26/42
Motivation Preliminaries Topological systems Systems & Spaces Problems Topological properties of spaces and systems
Definition 28 Given an algebra A and a subset S ⊆ A, S denotes the smallest subalgebra of A which contains S. Given a Q-space (X, τ), a subset S ⊆ QX is a subbasis of τ provided that τ = S. Lemma 29 The concrete category (LoA-Top, | − |) is topological. Proof. Given a | − |-structured source S = ((X, A)
(fi,ϕi)
− − − → |(Xi, Ai, τi)|)i∈I, the initial structure on (X, A) can be defined by τ =
i∈I Si with
Si = ((fi, ϕi)←)→(τi).
Variable-basis topological systems Sergejs Solovjovs University of Latvia 27/42
Motivation Preliminaries Topological systems Systems & Spaces Problems Topological properties of spaces and systems
Theorem 30 The concrete category (LoA-TopSys, | − |) is topological iff | − | is an isomorphism. Proof. Since the sufficiency is clear show the necessity. Let A
ϕ
− → B be a
Variable-basis topological systems Sergejs Solovjovs University of Latvia 28/42
Motivation Preliminaries Topological systems Systems & Spaces Problems Topological properties of spaces and systems
ϕ is injective For the singleton 1 = {⋆} define 1 × A
| =2
− − → A by | =2(⋆, a) = a. Let (1, A, B, | =1)
(11,1A,ϕop)
− − − − − − − → (1, A, A, | =2) be an initial lift of (1, A, B)
(11,1A,ϕop)
− − − − − − − → |(1, A, A, | =2)|. Given a ∈ A, a = | =2(⋆, a) = | =1(⋆, ϕ(a)) = | =1(⋆, −) ◦ ϕ(a). ϕ is surjective Define 1 × B
| =1
− − → B by | =1(⋆, b) = b. Let (1, B, B, | =1)
(11,ϕop,1B)
− − − − − − − → (1, A, B, | =2) be a final lift of |(1, B, B, | =1)|
(11,ϕop,1B)
− − − − − − − → (1, A, B). Given b ∈ B, b = | =1(⋆, b) = ϕ(| =2(⋆, b)) = ϕ ◦ | =2(⋆, −)(b).
Variable-basis topological systems Sergejs Solovjovs University of Latvia 29/42
Motivation Preliminaries Topological systems Systems & Spaces Problems Topological properties of spaces and systems
Proof. If A has the empty algebra A, then for every algebra B, the projection A × B
πB
− → B is an isomorphism and then B is the empty algebra. Thus Ob(A) = {the empty algebra}. If A has a non-empty algebra A, then for every algebra B, the projection A × B
πA
− → A is an isomorphism and then B is a singleton algebra. Thus Ob(A) = {singleton algebras}. In both cases A is a thin, connected category. It follows that the forgetful functor LoA-TopSys
|−|
− − → Set × LoA × LoA is an isomorphism. Q.E.D.
Variable-basis topological systems Sergejs Solovjovs University of Latvia 30/42
Motivation Preliminaries Topological systems Systems & Spaces Problems Topological properties of spaces and systems
Lemma 31 There is an embedding LoA-Top
E
Set × LoA × LoA
with E((X, A, τ)
(f ,ϕ)
− − − → (Y , B, σ)) = (X, A, τ)
(f ,ϕ,((f ,ϕ)←)op)
− − − − − − − − − − → (Y , B, σ). There is the projection Set × LoA × LoA Π − → Set × LoA with Π((X, A, B)
(f ,ϕ,ψ)
− − − − → (Y , C, D)) = (X, A)
(f ,ϕ)
− − − → (Y , C). The following diagram commutes: LoA-Top
(topological) |−|
|−| (non-topological)
Set × LoA × LoA
Π
Sergejs Solovjovs University of Latvia 31/42
Motivation Preliminaries Topological systems Systems & Spaces Problems Products and coproducts of systems
Suppose A has coproducts. Let ((Xi, Ai, Bi, | =i))i∈I be a set-indexed family of systems.
Let (X, (πi)i∈I) be a product of (Xi)i∈I in Set. Let ((µi)i∈I, A), ((ρi)i∈I, B) be coproducts of (Ai)i∈I, (Bi)i∈I in A.
Define X × B
| =
− → A with | =(x, −) =
i∈I
| =i(πi(x), −) given by the commutativity for every i ∈ I of the following diagram: Bi
| =i(πi(x),−)
B
| =i(πi(x),−)
µi
A
((X, A, B, | =), ((πi, µop
i , ρop i ))i∈I) is a (concrete) product of
((Xi, Ai, Bi, | =i))i∈I in LoA-TopSys.
Variable-basis topological systems Sergejs Solovjovs University of Latvia 32/42
Motivation Preliminaries Topological systems Systems & Spaces Problems Products and coproducts of systems
In general LoA-TopSys does not have concrete coproducts. It is still unclear whether LoA-TopSys has coproducts. Q-TopSys has concrete coproducts constructed as follows: Let ((Xi, Q, Bi, | =i))i∈I be a set-indexed family of systems.
Let ((µi)i∈I, X) be a coproduct of (Xi)i∈I in Set. Let (B, (πi)i∈I) be a product of (Bi)i∈I in A.
Define X × B
| =
− → Q by | =(x, b) = | =i(xi, bi) for x = µi(xi). (((µi, 1Q, πop
i ))i∈I, (X, Q, B, |
=)) is a (concrete) coproduct of ((Xi, Q, Bi, | =i))i∈I in Q-TopSys.
Variable-basis topological systems Sergejs Solovjovs University of Latvia 33/42
Motivation Preliminaries Topological systems Systems & Spaces Problems Problem 1
Lemma 32 LoA-TopSys
|−|
− − → Set × LoA × LoA creates isomorphisms. Lemma 32 and topological properties of LoA-TopSys suggest the following questions: Problem 33 Is the category LoA-TopSys algebraic? Does LoA-TopSys
|−|
− − → Set × LoA × LoA have a left adjoint?
Variable-basis topological systems Sergejs Solovjovs University of Latvia 34/42
Motivation Preliminaries Topological systems Systems & Spaces Problems Problem 2
By Lemma 23 LoA can be embedded into LoA-TopSys. Problem 34 Is it possible to embed LoA × LoA fully into LoA-TopSys? A possible answer It is possible to change the definition of LoA-TopSys in such a way that A × LoA can be fully embedded into it. The embedding has a left-adjoint-left-inverse Loc defined appropriately.
Variable-basis topological systems Sergejs Solovjovs University of Latvia 35/42
Motivation Preliminaries Topological systems Systems & Spaces Problems
amek, H. Herrlich, and G. E. Strecker, Abstract and Concrete Categories: the Joy of Cats, Repr. Theory Appl.
13 - 17, 1999. Coimbra: Univer. de Coimbra, Depart. de Matem´
Variable-basis topological systems Sergejs Solovjovs University of Latvia 36/42
Motivation Preliminaries Topological systems Systems & Spaces Problems
5–32.
paratopologies., Semin. de Topologie et de Geometrie differentielle Ch. Ehresmann 1 (1957/58), No.1, p. 1-9, 1959.
Variable-basis topological systems Sergejs Solovjovs University of Latvia 37/42
Motivation Preliminaries Topological systems Systems & Spaces Problems
145–174.
Verlag Dr. M¨ uller, 2008.
Variable-basis topological systems Sergejs Solovjovs University of Latvia 38/42
Motivation Preliminaries Topological systems Systems & Spaces Problems
24 (1968), 182–190.
between lattice-valued topology and topological systems, submitted to Quaest. Math.
Sostak, Axiomatic Foundations of Fixed-Basis Fuzzy Topology, Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory (U. H¨
Rodabaugh, eds.), Kluwer Acad. Publ., 1999, pp. 123–272.
Variable-basis topological systems Sergejs Solovjovs University of Latvia 39/42
Motivation Preliminaries Topological systems Systems & Spaces Problems
Fuzzy Topology, Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory (U. H¨
Kluwer Acad. Publ., 1999, pp. 273–388.
powerset theories and fuzzy topological theories for lattice-valued mathematics, Int. J. Math. Math. Sci. 2007 (2007), Article ID 43645, 71 pages, doi:10.1155/2007/43645.
Variable-basis topological systems Sergejs Solovjovs University of Latvia 40/42
Motivation Preliminaries Topological systems Systems & Spaces Problems
and spatiality, submitted to Proc. of International Conference
appear in Fuzzy Sets Syst.
variable-basis topological spaces, submitted to Soft Comput.
Variable-basis topological systems Sergejs Solovjovs University of Latvia 41/42
Motivation Preliminaries Topological systems Systems & Spaces Problems
Variable-basis topological systems Sergejs Solovjovs University of Latvia 42/42