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

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

2 Victor Porton

See my book for formulas for principal funcoids that is funcoids corresponding to binary relations

and for funcoids corresponding to pretopologies and preclosures (particularly funcoids corresponding

to topological spaces, as topological spaces can be considered as a special case of either pretopologies

or preclosures).

There are also several other equivalent ways to deﬁne funcoids.

Funcoids are made more interesting than topological spaces by a new operation (missing in tradi-

tional general topology), composition, which is deﬁned by the formula

(B, C, α

2

, β

2

) ◦ (A, B, α

1

, β

1

) = (A, C, α

2

◦ α

1

, β

1

◦ β

2

).

10. Reloids

Reloids are basically just ﬁlters on cartesian product A × B of two given sets A and B.

Formally:

Deﬁnition 11. A reloid is a triple (A, B, F ) where A, B are sets and F is a ﬁlter on A × B.

Reloids are a generalization of uniform spaces and of binary relations.

Deﬁnition 12. Composition g ◦ f of reloids can be easily deﬁned as the reloid determined by the ﬁlter

base consisting of compositions of binary relations deﬁning these reloids (see the book for an exact

formula).

Proposition 13. Composition of reloids (and of funcoids) is associative.

The sets of funcoids and reloids constitute complete atomistic co-brouwerian lattices and these

lattices have interesting properties.

There are interesting relationships between funcoids and reloids, as well as special classes of funcoids

and reloids.

14. Continuity

A function f from a space µ to a space ν is continuous iﬀ f ◦ µ v ν ◦ f. This formula works for

continuity, proximal continuity, uniform continuity, etc., so making all kinds of continuity described by

the same formula.

15. Other

My book [1] also considers pointfree generalizations of funcoids, multidimensional generalizations of

funcoids and reloids and other research topics.

I also introduce generalized limit for arbitrary (not necessarily continuous) functions.

My work introduced many new conjectures. So I give you a work.

Rerefences

[1] Victor Porton. Algebraic General Topology. Volume 1.

http://www.mathematics21.org/algebraic-general-topology.html