**Algebraic General Topology. Vol 1**:
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**Axiomatic Theory of Formulas**:
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Remark 1. It is a very rough partial draft. It is meant to express a rough research idea, not to

be correct, readable, or complete. First read the book:

http://www.mathematics21.o rg/algebraic-gene ral-topology.html upon which this formalistic is

based (especially about the deﬁnition of generalized limit).

See also http://planetmath.org/MetasingularNumbers

The idea is simple (for these who know funcoids theory). But to ﬁ nd exact formulations abo ut thi s

is notoriously diﬃcult. Below are attempts to formulate things about the theory of singularit ie s.

New theory

Deﬁnition 2. Singularity level is a transitive, T

2

-separable endofuncoid.

Let ν be a singularity level. Let ∆ be a ﬁlter.

Deﬁne SLA(ν) as:

Ob SLA(ν) = {ν ◦ f | f is a monovalued funcoid with domain ∆}

X [SLA(ν)]

∗

Y ⇔ ∃x ∈ X ∀K ∈ GR x∃L ∈ Y : L ⊑ K

[FIXME: It is probably not a funcoid .]

Remark 3. GR x is used despite of it is a funcoid not reloid.

Proposition 4. SLA(ν) is an endofuncoid.

Proof. ¬(∅ [SLA(ν)]

∗

Y ) and ¬(X [SLA (ν)]

∗

∅) are obvious.

I ∪ J [SLA(ν)]

∗

Y ⇔ I [SLA(ν)]

∗

Y ∨ J [SLA(ν)]

∗

Y is obvious.

X [SLA(ν)]

∗

I ∪ J ⇔ ∃x ∈ X ∀K ∈ GR x ∃L ∈ I ∪ J: L ⊑ K ⇔ ∃x ∈ X ∀K ∈ GR x :

(∃L ∈ I: L ⊑ K ∨ ∃L ∈ J: L ⊑ K)??

??

Alternative deﬁnition:

[FIXME: It is probably not a funco id.]

Deﬁnition 5. X [SLA(ν)]

∗

Y ⇔ ∃z ∈ Ob µ∀K ∈ GR z ∃x ∈ X , y ∈ Y : x, y ⊑ K

Proposition 6. SLA(ν) is a funcoid.

Proof. X [SLA(ν)]

∗

Y ⇔ ∃z ∈ Ob µ , x ∈ X, y ∈ Y ∀K ∈ (GR z)

X ×Y

: x, y ⊑ K

x,y

⇔ ??

I ∪ J [SLA(ν)]

∗

Y ⇔ ∃z ∈ Ob µ∀K ∈ GR z ∃x ∈ I ∪ J , y ∈ Y : x, y ⊑ K ⇔ ∃z ∈ Ob µ ∀K ∈ GR z:

( ∃y ∈ Y : y ⊑ K ∧ (∃x ∈ I: x ⊑ K ∨ ∃x ∈ J : x ⊑ K)) ⇔ ∃z ∈ Ob µ∀K ∈ GR z: (( ∃y ∈ Y : y ⊑ K ∧ ∃x ∈ I:

x ⊑ K) ∨ ( ∃y ∈ Y : y ⊑ K ∧ ∃x ∈ J: x ⊑ K)) ⇔ ??

Proposition 7. SLA(ν) is T

2

-separable.

Proof. ??

Proposition 8. SLA(ν) is transitive.

1

Proof. ??

Galufuncoids

Let A and B are Rel-morphisms. I will denote like (∼

A

) = GR A and (∼

B

) = GR B.

Deﬁnition 9. Galufuncoi ds between A and B is a quadruple (A; B; α; β) such that

∀x ∈ Ob A, y ∈ Ob B: (αx ∼

B

y ⇔ x ∼

A

βy).

Deﬁnition 10. x [f] y ⇔ x ∼

Src f

βy.

Obvious 11. x [f ] y ⇔ x ∼

Src f

βy ⇔ αx ∼

Dst f

y.

Remark 12. Galuf uncoids are a gener alization of both (pointfree) funcoids and Galois connec -

tions.

Deﬁnition 13. The reverse galufuncoid is deﬁned by the formula:

(A; B; α; β)

−1

= (B; A; β; α).

Proposition 14. Composition of (composable) galufuncoids is a galufuncoid.

Proof. (α

2

◦ α

1

)x ∼ y ⇔ α

2

α

1

x ∼ y ⇔ α

1

x ∼ β

2

y ⇔ x ∼ β

1

β

2

y ⇔ x ∼ (β

1

◦ β

2

)y.

Obvious 15. Galufuncoids form a category (sim ilarly to the category of pointfree funcoids).

Deﬁnition 16. On the set of galufuncoids is deﬁned a preo rder by the formula: f ⊑ g ⇔ [f ]⊆[g].

Galufuncoidal product

Functional galufuncoid

Deﬁnition 17. Functional galufuncoid ν/∆ of ν thr ough ﬁlter ∆ is the endo-galufuncoid deﬁned

by the formulas:

Ob(ν/∆) = FCD(Base(∆); Ob ν)

hν/∆if = ν ◦ f and h(ν/∆)

−1

if = ν

−1

◦ f

f ∼

Ob(ν/∆)

g ⇔ g

−1

◦ f ⊒ id

∆

FCD

[TODO: Restric t to the special case f = ν ◦ F to make i t T

2

.]

[TODO: X [SLA(f )] Y is deﬁned as existence of x ∈ X such that for every entourage of x ther e is

y ∈ Y which is a subﬁlter of this entourage.]

Obvious 18. ∼

Ob(ν/∆)

is a symmetric relatio n.

Proposition 19. This is really a galufuncoid and f [ν/∆] g ⇔ g

−1

◦ ν ◦ f ⊒ id

∆

.

2

Proof. We need to prove

hν/∆if ∼

Ob(ν/∆)

g ⇔ g

−1

◦ ν ◦ f ⊒ id

∆

⇔ f ∼

Ob(ν/∆)

h(ν/∆)

−1

ig .

Really,

hν/∆if ∼

Ob(ν/∆)

g ⇔ ν ◦ f ∼

Ob(ν/∆)

g ⇔ g

−1

◦ ν ◦ f ⊒ id

∆

f ∼

Ob(ν/∆)

h(ν/∆)

−1

ig ⇔ f ∼

Ob(ν/∆)

ν

−1

◦ g ⇔ g

−1

◦ ν ◦ f ⊒ id

∆

Remark 20. A way to come to the above formula

∀x ∈ atoms ∆: fx [ν] gx ⇔ ∀x ∈ atoms ∆: x [g

−1

◦ ν ◦ f ] x ⇔ g

−1

◦ ν ◦ f ⊒ id

∆

.

Hierarchy of singularities

Consider two endo-galufuncoids µ and ν. Values on Ob µ will be have like arguments of functions,

of Ob ν like values of functions.

I call SLA(Ob µ) singularity level above Ob µ the set of sets of funcoids ν ◦ f |

hµi

∗

{x}

(or alternatively

of lim its xlim f |

hµi

∗

{x}

) where f is a monovalued principal funcoid in FCD(Ob µ; Ob ν).

[TODO:

Maybe exclude the zero funcoid?]

Consider a galufuncoid ω deﬁned by the formulas:

hωif = ν ◦ f and hω

−1

if = f ◦ ν and f [ω] g ⇔ ∃x ∈ Ob ν: g

−1

◦ f ⊒ id

hµi

∗

{x}

FCD

.

We need to prove

hωix ∼

ω

g

−1

⇔ ∃x ∈ Ob ν: g

−1

◦ (∼

ν

) ◦ f ⊒ id

hµi

∗

{x}

and ??

The ﬁrst is equivalent to (∼

ν

) ◦ f ∼

ω

g

−1

⇔ ∃x ∈ Ob ν: g

−1

◦ (∼

ν

) ◦ f ⊒ id

hµi

∗

{x}

FCD

.

Really, (∼

ν

) ◦ f ∼

ω

g

−1

⇔ ∃x ∈ Ob ν: g

−1

◦ (∼

ν

) ◦ f ⊒ id

hµi

∗

{x}

Lemma 21. Let ν ◦ ν ⊑ ν and ν

−1

◦ ν ⊑ ν. If x and y are ultraﬁ lters, then x [ν] y ⇔ hν ix = hν iy.

Proof.

[TODO: Prove fo r the more general case of galufuncoids. It’s problematic.]

x [ν] y ⇔ y ⊑ hν ix ⇒ hν iy ⊑ hν ◦ ν ix ⇒ hν iy ⊑ hν ix. So taking symmetry into account we have

x [ν] y ⇒ hν ix = hν iy.

Let now hν ix = hν iy, Then y

hν

−1

◦ ν ix; y ⊑ hν

−1

◦ ν ix; y ⊑ hν ix; y

hν ix; x [ν] y.

Theorem 22. f [ν/hµi

∗

{x}] g ⇔ ν ◦ f |

hµi

∗

{x}

=ν ◦ g |

hµi

∗

{x}

for f , g ∈ SLA(Ob µ).

[TODO:

Generalize it for any ∆ instead of hµi

∗

{x}?]

Proof. It’s enough to prove hf ix [ν] hgix ⇔ hν ihf ix = hν ihgix for every ultraﬁlter x.

hf ix [ν] hg ix ⇔ hgix ∼

Ob µ

hν ihf ix ⇒ ?? ⇒ hν ihgix ∼

Ob µ

hν ihf i

??

Theorem 23. SLA(Ob µ) is T

2

-separable .

The reloid ν

′

on the set SLA(Ob ν) could be deﬁned by one of the two formulas below:

3

For every point x get

σ

f

(x) =

u ∈

[

f | dom u = hµi

∗

{x}

.

q

f

=

F

x∈Ob µ

σ

f

(x).

f [ν

′

] g ⇔ ∃x ∈ Ob µ:

G

q

f

[ν/hµi

∗

{x}]

G

q

g

.

or

f [ν

′

] g ⇔ ν ◦ q

f

= ν ◦ q

g

.

[TODO: How to deﬁne galufuncoids corresponding to the above formulas (if at all possible)?

∃x ∈ D: f [ν(x)] g ⇔ ∃x ∈ D: g

hν(x)if ⇔ ?? ⇔ g

F

x∈D

ν(x).

The ?? does not generally hold: our lattices are co-brouwerian not brouwerian!

]

The above formula holds if g is a discrete reloids. So repla ce every funcoid f ∈ SLA(Ob ν) with

(RLD)

in

f. Then continue for arbitrary reloids.

Another try: hν

′

ix = ν ◦

FS

x

y

hν

′

ix ⇔ y

ν ◦

FS

x ⇔ ν

y ◦ (

FS

x)

−1

⇔ ν

−1

(

FS

x) ◦ y

−1

[FIXME: x and y are of

diﬀerent types.]

x

hν

−1

iy ⇔ x

ν

−1

◦

FS

y

The rest

One more ot he r deﬁnition:

f [ν

′′

] g ⇔

G[

g

−1

◦ ν ◦

G[

f

0

Yahoo! (i ∪ j) [ν

′′

] g ⇔ i [ν

′′

] g ∨ j [ν

′′

] g etc.

Proof. (i ∪ j) [ν

′′

] g ⇔ (

FS

g)

−1

◦ν ◦ (

FS

(i ∪ j))

0 ⇔ (

FS

g)

−1

◦ν ◦ ((

FS

i) ⊔ (

FS

j))

0 ⇔

(

F S

g)

−1

◦ ν ◦ (

F S

i) ⊔ (

F S

g)

−1

◦ ν ◦ (

F S

j)

0 ⇔ (

F S

g)

−1

◦ ν ◦ (

F S

i)

0 ∨

(

FS

g)

−1

◦ ν ◦ (

FS

j)

0 ⇔ i [ν

′′

] g ∨ j [ν

′′

] g

Proposition 24. ν

′′

is a galufuncoid.

Proof. ??

An attempt of an alternate deﬁnition:

f [ν

∗

] g ⇔ xlim f ◦ ν ◦ xlim g

0 [FIXME: Does this make sense?] [TODO: Diﬀere ntiate generalized

limit as a set of funcoids or its variation as a funcoid-value function.]

Proposition 25. xlim f ∈ SLA(Ob ν) if f ∈ FCD(Ob µ; O b ν) \

0

FCD(Ob µ;Ob ν)

.

Proof. ??

Proposition 26. τ (x) ∈ SLA(O b ν).

Proof. ??

4

Proposition 27. τ (x) [ν

′

] τ (y) ⇔ x [ν] y.

Proof. ??

Metasingular numbers

Let y ∈ SLA(Ob µ). I will denote r(y) such x ∈ Ob ν that τ (x) = y, if such x exis ts.

I will call base singular numbers the set BSN = Ob ν ∪ SLA(Ob ν) ∪ SLA(SLA(Ob ν)) ∪

.

[TODO: Set that this union is di sjoint.]

I will call meta-singular numbers the set MSN = {y ∈ SLA(Ob ν) | ∄x ∈ BSN: y = τ (x)}.

Deﬁnition 28. I call reduced BSN its corresponding MSN (that is r applied to our BSN a natural

number of times while possible).

Deﬁnition 29. I call reduced limit the reduced generalized limit.

Functions with meta-sin gular numbers as arguments

Let f is an n-ary (n is an arbitrary possibly inﬁnite index set) function o n O b ν. Then deﬁne

function f

′

on SLA(Ob ν) as:

f

′

(b) =

(

g ◦

Y

(A)

b | g ∈ f

′

)

.

We c an’t use cross-composition product instead of above sub-atomic product because cross -compo-

sition product is not a funcoid (just a pointfree fun coid). We can replace sub-atomic product with

displaced product, but as about my opinion displaced product seems more weird an inconvenient.

The above ind uc e s a trivial deﬁnition of functions on MSN but only for functions of ﬁnite arity

(because having a ﬁnite set of MSN we can raise them to the same (maximum) level).

On diﬀerential equations

Replacing limit in the deﬁnition of derivative with the a bove deﬁned reduced limit, the base set

Ob µ with MSN and operations f on the set Ob µ with corresponding operations on MSN, we get

a new interpretation of a diﬀerential equation (DE) (ordinary or partial).

Let call such (enhance d) diﬀerential e quations m eta-singular equations (as opposed to non-singular

equations that is customary diﬀerential equations).

There arise the following questions:

Deﬁnition 30. I call a solution of a DE a trivia l restriction if it is a restr ic tio n (to t he set of non-

singular points) of exactly one enhanced DE.

We need to ﬁnd when there are solutions which are not trivial restrictions.

Then we can split such non-trivial solutions into follow ing classes:

• “added solutions” are solutions whose restriction t o non-singularity points is not a non-

singular solution;

5

• “alternate solutions” is when an non-singular solution is a restriction of more than one meta-

singular solution;

• “disappearing solutions” when a non-singular solution is not a restriction of a meta-singular

solution.

Special case of general relativity

I am not a expert in general relativity (I am not even a professional mathematician).

But it looks like that the equations of general relativity can be converted (as described above) into

meta-singular equations. For the special case of general relativity equa tions, the above classes are:

• “added solutions” would possibly characterize a “world above” described not with real num-

bers as our world but with singularities. This may or may not be of physical interest.

• “alternate solutions” would characterize black (or white) holes with additional information

hidden inside. This a dditional information may probably solve the well k nown pa radox of

information disappearing when it fa lls into a black hole.

• “disappearing solutions” would mean that the laws of nature are possibly more restrictive

than c onsidered in more traditional physics. Could it resolve time-machine related p ara-

doxes?

I again repeat that I am not an expert in general relativity. I seek collaboration w ith general

relativity experts to solve the problems I’ve formulated.

I think (except of the case of the negative result that is there are no non-trivial solutions) this

research is destined to receive Nobe l Prize and/or Fundamental Physics Prize. I want my half.

Note that the g roup G (see the deﬁnition of gene ralized limit in my book) for general relativity

can be deﬁned in two diﬀerent ways: as the group of homeomo rpisms of the curved space or as the

group of only uniformly continuous (in both directions) bijections. This gives us two new theories

of general relativity.

6