
Algebraic General Topology (theory of funcoids and reloids)
Victor Porton
(Shay Agnon 32-29, Ashkelon, Israel)
E-mail: porton@narod.ru
In my book [1] I introduce some new concepts generalizing general topology, including funcoids and
reloids. The book along with supplementary materials (such as a partial draft of the second volume)
is freely available (including the L
A
T
E
X source) online.
Before studying funcoids, in my book I consider co-brouwerian lattices and lattices of filters in
particular, as the theory of funcoids is based on theory of filters. My book contains probably the best
(and most detailed) published overview of properties of filters.
Because you always can refer to my book, in this short intro I present theorems without proofs.
I denote order on a poset as v and corresponding lattice operations as
F
and
d
. I denote the least
and greatest elements (if they exist) of our poset as ⊥ and > correspondingly.
1. Filters
Definition 2. I order the set F of filters (including the improper filter) reverse to set-theoretic order,
that is A v B ⇔ A ⊇ B for A, B ∈ F.
Proposition 3. This makes the set of filters on a set into a co-brouwerian (and thus distributive)
lattice, that is we have A t
d
S =
d
X ∈S
(A t X ) for a set S of filters and a filter A.
4. Funcoids
Let F(A), F(B) be sets of filters on sets A, B. They are complete atomistic co-brouwerian lattices.
Definition 5. A funcoid A → B is a quadruple (A, B, α, β) where α and β are functions F(A) → F(B)
and F(B) → F(A) correspondingly, such that Y u α(X ) 6= ⊥ ⇔ X u β(Y) 6= ⊥ for every X ∈ F(A),
Y ∈ F(B).
Definition 6. I denote (A, B, α, β)
−1
= (B, A, β, α).
Definition 7. I denote h(A, B, α, β)i = α.
Funcoids generalize such things as:
• binary relations;
• proximity spaces;
• pretopologies;
• preclosures.
For a proximity δ, define
X δ
0
Y ⇔ ∀X ∈ X , Y ∈ Y : X δ Y
for all filters X , Y. Then we have a unique funcoid f such that
X δ
0
Y ⇔ Y u hf iX 6= ⊥ ⇔ X u hf
−1
iY 6= ⊥.
Definition 8. X [f] Y ⇔ Y u hfiX 6= ⊥ ⇔ X u hf
−1
iY 6= ⊥.
Proposition 9. A funcoid f : A → B is uniquely determined by hfi and moreover is uniquely deter-
mined by values of the function hf i on principal filters or by the relation [f] between principal filters.
1