IC/T9/120INTERNAL REPORT

(Limited distribution)

International Atomic Energy Agency

and

United Nations Educational Scientific and Cultural Organization

INTERNATIONAL CENTRE TOR THEORETICAL PHYSICS

QUASI-CATASTROPHES AS A NON-STANDARD MODEL

AND CHANGES OF TOPOLOGY •

F. Destefano

Scuola dl Perfezlonamento in Matematica, Universita dl Trieste, Italy,

and

K. Tahir Shah ••

International Centre for Theoretical Physics, Trieste, Italy.

MIRAMAKE - TRIESTE

August 1979

• To be submitted for publication.

** On leave of absence from Department of Mathematics, Alfateh University,Tripoli, Libya.

ABSTRACT

The concept of a singularity is clarified using a non-standard model

of mathematics. Also ve show that there exist a relationship between the

existence of a singularity and non-homeotnorphie change in topology. To verify

our theory and to develop a mathematical formalism (a non-standard one) we have

worked out various topologies on non-standard real line. This vill be

developed later on to include theory of curves, surfaces and manifold. Our

non-standard analysis differs from that of Robinson and represents a step

further in generalization.

-1-

I. INTRODUCTIOK

This mathematical exercise developed out of a search for a suitable

mathematical machinery vhich can be utilised to define in a unique and general

fashion the singular structure of a physical theory. There are as many

definitions of. singularity as one can find examples in physical theories and

mathematics. One vould, therefore, conclude that there is no definite consensus

among physicists or mathematicians as to what one shall call a singularity

vhich fits into all situations. We have taken a first step concerning this2a)

problem recently by introducing the concept of quasi-catastrophe vhich we

shall utilize in this paper. In another note we have discussed some logico-

mathematical aspects of singularities in physical theories and it is proposed

to use the logic off variable predicate calculus. However, this suggestion

although philosophically significant cannot be of much help in practical

calculations. To fill this gap ve shall extend our ideas to more practical

aspects and apply them to real analysis and topology of manifold to be eventually

used in physical theories. The first part of this mathematical programme (this

paper) is to study the non-standard model of real line from a topological

point of view.

Going through an immense amount of literature on the questions related

to singularities in various contexts, one comes to the conclusion that there

are two basic questions in this regard. What is the meaning of a singularity?

What is the "boundary" of a relevant topological apace or a differentiable

manifold? More fundamentally, what is the boundary of a set (relevant to

the problem concerned) and hov it is related to the "singularity" if at all

it can be defined properly? These are the basic questions ve have in mind

and in the process of answering these, ve come to the development of a new

non-standard model. We shall start by posing these questions for real line

and then go over to more complicated cases. This model differs fromIt)

non-standard analysis in its basic philosophy and results

' ' of constructing

Robinson's

obtained. It also differs from various other methods

boundary but it has some similarities with the super-extensions approach

developed by Verbeck and de Groots et al. in the context of topological

spaces. We have also borrowed the idea of variable quantities and variable

7)structures in Topoi from Lawvere and used it in the formulation of the

quasl-eatastrophe theory.

One can easily note that there is one common aspect of all types of

singularities. It represents a breakdown of some property, condition, or

equation vith vhich ve identify the theory. For instance, we knov from

-2-

school mathematics that "infinity" is something which cannot be defined within

the real number system. It is "beyond" the real numbers. In fact, infinity

or infinitesimals are not very large or very small quantities in our view but

Bimply they do not belong to the class of reals. The concept of infinitely

large or small entities has apparently come through extrapolation of division

by fractions less than one "up to" zero. In our viev, this is not correct and

leads to confusion. Below we shall elaborate On this point also.

In Sec.II we give known preliminaries on non-standard models and

Sec.Ill describes the concept of quasi-catastrophe and its relation to

singularity. In Sec.IV we discuss in detail topologies on a non-standard

model of real line {which is not linear in the conventional sense!).

II. NON-STANDARD MODELS

A very large part of mathematics is concerned with the study of

"standard" mathematical systems such as natural numbers, the rationals, the

real and complex numbers etc., each of which is regarded as a- unique system.

One finds that within the first-order predicate calculus.axiomatization of

these systems cannot be categorical and there exist models of our axiomatic

theory not isomorphic to the system we wish to study. We exploit such models

to define and study the standard system with "singularities". Let us, therefore,

recall a few definitions.

Definition 2.1

An elementary standard system ^ is a set £ together with a subset

"Hi of the set of relations on such as "5 6 *3l, , where 3 I s t n e identity

relation symbol.

Example 2.1

The elementary number system 1ft, cons i s t s of t he se t tR, of r e a l

numbers toge the r with the s e t "St. = { 3 , + , X , < } of r e l a t i o n s on fR,

Here + is the ternary relation (x,y,z)£ + if and only if x + y = z etc.

Let ,& = (jS'jJl) t>e an elementary standard system. We take a set

V O/^' , such that V - iS' is countably infinite and form the first-order

algebra P^ ' ) = P(V,S.).

Definition 2.2

The language of ,JSf is the subset

-3-

interpreting each element of «S' and 3v as itself assigns a truth value

v{p) to each

Definition 2.3

The theory of

A = {p€iUT)/v(p) - 1

is the theory T(«S") = (ft, A,,X) where

ia a complete theory with , * as a model. The theorems of this

theory are its axioms and consist of all elements of sC If) which are true

in iS' or in any other model of T(*^). If the axiom set A were fully

known, then TCuf) could give us no new information about ,£' . However,

our knowledge of »S" is usually incomplete, and any method which extends our

knowledge of A in fact extends our knowledge of nS1 . Suppose now we

•choose a model tS1 • of T(»?) such that the truth or falsity of certain

statements P£oC(nST)is more easily determined (by arguments in the meta-

language) fariS' * than for , then we have a method of utilizing T(w?*)

to discover properties of „£" . Our aim is to construct such a model«!*•

and utilize it to define singularities but not in terms of incompleteness.

Def in i t ion 2.U

Let ,,S' • be any model of T(»?). We say that,? • is a standard model

of Tdtf ) if J? • is Isomorphic to ,5* , and otherwise „? • is called a

non-standard model of T(»? ).

Let,.?* = C^"«,f,6) be any model of T(»£) and f i^1-*^'* embeds

MS1 in ,&* such that if (a 4 b) € A then f(a.) f f(b). In general, for

any n-ary relation p ( 'Si, , the restriction to f W ) of the relation g(p)

is the relation on fUS1) which corresponds under f to the relation p on

Jf . We shall therefore mostly identify ^S* with its image under f in,;?*,

and so regard the model ,& * as containing the standard model

III. SINGULARITY AND QUASI-CATASTROPHE

l) 2)Consider two (constant predicate) sets * S = {s.,s ,...,s ,...}v £. n

and S = {s ,S2,...,3 , } each specified with respect to properties P and

P respectively. In other words, all elements {s.,vi ™ 1,2,,,. ,n,...} are

equivalent with respect to property P P and all elements iBjs 1 = l>2,...,n,

are equivalent with respect to P. Here set means, e.g.,a set of topological

spaces, set of equivalent dynamical systems, or set of states of a physical

system and so on. The property P (respectively P) in essence, defines an

equivalence relation and the elements s. and s. (i J) are equivalent with

-k-

respect to P, I.e. having the same property. Intuitively, ve are tempted

to define a Jump from the set S to 3 under some kind of limiting

operation, so-called quasi-catastrophe to express the fact that a sequence of

elements of 3, called {s^} have a limit ifiS . This is general enough

to define a singularity tut to compare it with various versions of singularities

one must specify further the structures of the sets S and S. For Instance,

if S • 1R the real line with the usual structure (say topological), then a

sequence of real numbeismsy converge (conventionally it is called diverge) to

an element of the set {+*, - " } . Here set S = {+*, -»}. . This is the

traditional definition of a singularity. For us it is simply a passage from

one class to another. In physical theories we need some kind of metric space

so as to measure the values of physical entities such that values remain real

numbers. In the case that one gets the ansver infinity (never experimentally

but only theoretically) one says that the theory is singular! The problem

Is therefore not with the nature but with our mathematical system. In fact

the existence of the non-standard model demonstrates that this is the case.2)

This definition is meaningful and non-trivial results are obtained for

G-equivalence and also stability properties of the invariant theory (Hilbert-

Mumford) is an example vhieh fits well.

Coming back to the metric structure, we can define metric induced

non-equivalent topologies on S and S, say one continuous and the other

discrete, respectively. Then breakdown of continuity and differentiability,

the two basic properties, is simply due to change of topology. In other words,g)

if the equivalence relation is topological equivalence , then singularity

is associated vith the (non-homeomorphlc) change of topology. Similarly,

taking into account homotopy equivalence, change of homotopy type results in

a singularity or breaking of so-called topological conservation lews recently

discovered by physicists.

The topological equivalence relation is extremely important because

the fundamental notions of limit, continuity and differentiability are "11

related to it. As we know all physical theories, and to some extent standard

analysis and geometry depend heavily on the concept of differentiability (a

linear concept) or smoothness, one is forced to conclude that the occurrence

of a "singularity" and breakdown of differential equations representing laws

of nature at some value of a parameter of the theory, are a. result of topology

change in an abrupt manner not desirable by standard analysis. Consequently,

ve consider change of topology as a fundsmental mechanism responsible for

the occurrence of singularities.

- 5 -

Comparing this with the non-standard analysis of Robinson we realise

that our approach, seems a bit more general mathematically and obviously close

to physical reality.

IV. TOPOLOGIES OH A HOH-STAHBABD MODEL OF REAL LIITE

k.l . •• S e a l l i n e i s the simplest t opo log i ca l space with r ich enough structure

t o be considered as a prime candidate for t e s t i n g our theory, and not only

t h i s but p h y s i c i s t s are w e l l accustomed t o i t . Results o f experiments are

real numbers and a l s o t h e o r e t i c a l predict ions are always i n terms of rea l

numbers t o be obtained by measurement. For these simple but fundamental

reasons ve begin our programme of study with one-dimensional Euclidean rea l

topological space called real l ine.

Let us consider (%,d) one-dimensional Euclidean space fU. with metric

d. Let us consider the set B • {+*>, -«•} , We then define

d : ' B « B + at where H* • { i | x « B , x %. 0}

in the following manner:

*{+., - ) - d ( - , - ) - 0

a(+-, _») = a(-», +-)€(H+ "here (R.+ - (xjxi m, x > 0)

3 is then a metric (in the conventional sense) on B and satisfies all the

axicas of a metric (see, e.g. Dleudonne1, Ref.8). It la well known that, if

(X,d) is a metric space (for any set X and metric d) and if ve take the

open set

p£ = <y|d(x,y) < r}

JP"1 •then the collection f *J r*t forme a basis for the metric topology of X.

Consequently, one can see m a n the above definition that the topology on the

metric apace (B,d) is a discrete topology. In fact, the open balls are

({+•} if r 3 d(+-, -«)

{+», — } . B if r > a(+-, — )

r /{--} if r & d(+-, -«>)

M+-, —) if r > d(+-, — )

(U.la)

Let us noy take the aet 51* = |B u B . One would like to see now

whether i t Is possible to deftne a distance function d* on flfl guch that

d*| = d ( i . e . res t r ic t ion to IR. coincides with Euclidean metric)

(It.2a)

= d (i.e. restriction to B coincides with 3) . (k.2b)B x B

How, we are confronted vith the problem as to hov we can define a function

W. ->fR- which associates to every point of W a positive number which

iB to be called distance from that point to +• {respectively, — » ) , i.e.

(Here distance is from a fixed point +«° (respectively, -») to a variable

point.) such that

d«(•, +«) = d*(+-, •) and d»(-, -») » d«(-», •) .

We assume that all the axioms of the distance function are valid for

d« :%* x m « + m * , i.e.

1) d«(x,y) > 0 if x t y Vx,y«IR»

2) d»(x,y) - 0 if x - y

3) d«(x,y) - d»(y,x)

1*) a»(x,y) d»(x,z) + d*(z,y) V x,y,z *|R» . (I,.3)

We shall see in the sequel tha t , while axioms 1, 2 and 3 (Eq.(h.3)) are

satisfied by d», i t is impossible for d# to satisfy axiom h, tar the

cases we have investigated below. This is not only due to the fact that we

want +• and -•» to he adherent pointefor points of accumulation vith the

usual meaning, i . e . we want a l l the infinitesimaly large (respectively small)

points to be "near" +- (respectively, -HS, to [R. in 0t», but also i f we simply

want that d"(x, +~) be a non-increasing function of x, or that d*(x, -~)

be a non-decreasing function of x.

Let d« : m» x K» + |R* satisfy metric axioms 1, 2 and 3 (Ea..(!*.3))and that conditions (U.2a) and Ct.21)J holds t rue . Furthermore,

a . 3 M, such that for every x,yeK, x.,y > M,

impliesx > y - —fr d«U, +-) ^ d*(y, +«)

such that for every x.ytfR, x,y <

. — ) •

b .

Thesis (U.I)

d* is not a distance function in the usual sense. If axiom h of

Eq.(l*.3) is imposed on all points of IB.*, then it leads to contradiction.

Proof

Let us suppose tbat axiom 1* (Eq.(l*.3)) is verified. Taking i,y« (R

such that x > Mj and x > 0, we take x >. fy| . Writing the triangular

inequality for the ternary (x,y,+~)

d»(x,y) 4 d«(x, +-) + d«{y, +-) .

We now consider the following point of R:

z = x + d»(x, +») + d*(y, +•) ,

which sat isf ies z > 1 and hence z > I t . Therefore,

A*(z, +«) ^ d»(x, +«) (hypothesis U.la^) .

The triangular inequality for ternary (i,y,+«) Is

d»(a,y) 4 d»(z, +-) + d»(y, +-)

and from the condition d # L « d, we get|iR. x R

|z-y| « d»(z, +-) + d»(y, +-) f d»(x, +-) + d»(y, +«) (from hypothesis U.l^

While using the definition of the point z, one has

|x + d*U, +«•) + d»(y, +-) - y| 4 d«(x, +») + d»(y, +-)

- 7 - -fl-

and therefore \x + d»tx, +-) * d«(y, +"0| - |y| i d»(x, +-) + &•(*, +-)•Here ve use the relation |al - \t>\ i !a-b[ which ta certainly valid forpoints belonging to R. Since x, d*(x, +-) and d»(y, +-) are positive,one finds after simplification x - |y[ rf O = ^ x . J )y| , which contradict*the in i t ia l choice of y. For hypothesis It.lb. one can prove analogously.

It does not come aa a great surprise to us that the triangularinequality is not aatiafied for IR« = BoB. The triangular inequality iastrict ly a property of linear geometry and one knows that usually the geometryof the boundary of a manifold is not of the same type as the manifold andhence there la no reason to believe that the boundary of an Euclidean spaceis also Euclidean. We do not inBist therefore on having our metric d» thetriangular inequality aatisfied, instead we care about having i t s restrictionto IR behaving as the usual Euclidean netric.

To illustrate further what kind of metric dKve may call i t a. non-standard metric) is,we give a few examples below.

Examples

1. Choose the following values for d»:

d*(x, +-) •> h and d»{x, -»)

x » h, y = -2h, one obtains 3h =d*{y, +•*) = 2h contradiction.

It for every x£(&, and let|x-y| » d»(x,y) « d»(x, +») +

Conaidering this d» as a new type of distance (non-standard) and

considering the definitions of the open ball*,- one concludes that +» and

-» are isolated points of (HP.

2. Let us now take

d»(x, +•>) - e~x and d»(jc, --) - e x V i t I R

again the inequality d«(x, -») « d«{x,y) + d"(y, -») leads to contradiction.

In this case +» and -" should have been adherent points of IR, i.e. if

we call the open sets the open balls of centre +•> or -"> and radius r«(R ,

then in eveiy open ball there »re points of IR.

xClB : e"1 < r. Furthermore, if we chooee

r > a»(+», -") * d(+», - " ) , then also

-9-

la a part of

3. Let us put

d»U, +-] - 3- _ arctg x V

d»(x, - -) - j * arctg x ,

and d»(+-, -») = it * d(+», -«) . We would like to verify the inequality

d*(x,y) ^ d*(x, +») + d*(y, +»), x.yefR.

la |x-y] 4 •£ - arctg x + j - arctg y. Posing x = y + 3* , we get thecontradiction.

Comparing carefully the metrization procedure' for JW followed byDieudonne , we note that in his case the condition d* L x ^ * d i s notsatisfied. Given H, I = (-1, +1) the open interval, f; fil + I : f(x) • f ithen f i s a bijection and Its inverse ia g(y) » (f~ Cy)) * i' i 'T • Puttingnow J • [-1, +1], one can define g : J + IK1 • Illu {»} u { - " } , such that

gCl) = +" , i ( - l ) " -» and f « g~ . As a conclusion ?|_ • f,+») • 1 and f(-») = - 1 . Since J ia a metric space with respect to

the Euclidean distance [x-y| , tR* ia metrlsed by carrying the distance of Jon DR» via mapping f by puttingd(x,y) = ^ o i i ^ a n ^ f x K *(y) " |?(x)-fry)|V x,y cIR* . One realizes that this distance for two points in IR iadifferent from the Euclidean one. In fact, supposing |x-y| " |f(x) - ?(y)|

one gets false equality

g

f(

k.2 Since It 1B not possible to give d* al l the usual properties of &metric, we ahall now explore the poBBibllity of defining a topology x on (R»in such a manner that topology Induced on the subset R JR* by the topologyT Is a Euclidean one and that on subset BelR* a discrete one. In generaltopology on B should be different frcm that on IR induced by T. Thegeneralization of this particular problem is as follows. Given two topologicalspaces UifA ) and (B.tg), give a topology v on AuB such that the topologiesinduced on A and that on B coincides, respectively, with the in i t ia ltopologies T and x (or at least homeomorphlc to T, and x_, respectively).A a A ' BIn our case the discussion of the problem is facilitated by the fact that

Apparently there are tvo solutions to this problem. The first is a-

non-intuitive approach considering infinity not as a "very large" or "very

small" entity but something simply out of the real numbers system. The second

-10-

ia a conventional approach, considering infinity as something "extremely large"

(small, respectively). In the first case we can define open sets of the

topology on 1RUB in the following manner.

Given the open balls P with a real point as centre and positive

radius r, we can call open the following sets:

ftxlR.

IB*BI. C , cL" = 0

Also

and also their arbitrary unions and finite intersections. It is obvious that 4>

d, are open sets. The open sets of the induced topology on P a r e then (by

definition) the intersections of PL with the open sets of (JR*,T), i.e. the

arbitrary unions and finite intersections of the tails P* , since +«• , -»^|

One can verify this for each case, for instance

and

-- QP;

In a similar fashion the topology induced on B by T is the discrete

P r has elements in B . One should notetopology,in fact none of the 'balls P r has elements in

that the topology X is a Hausdorff topology.

due to the fact that

Also IR." is not connected

d» satisfies axioms 1,2,3 (Eq.(U-3)) hut not axiom k, in general, unless the

ternary elements of R* are chosen in a particular manner. In fact i f

x,y,z{(A, then axiom U holds true, i . e . |x-y| ^ |x-i | + |z-y[, and also i f

we take aft element xtft and two elements of B. One has for instance

1. d«(x, +«) * d»(x, —) + d»{+«,-») since - ~f arctg x is true

for every xfetR;

2. d»(x, --) < d»(x, +•) + d»(—>,+«) since arctg x * ~ V*£ (Rj

3. d»(+-,—) t d*(+«, x) * d»(x, —) .

Let us define as open balls with centre +«O {-co, respectively) and

radius r > 0 the following sets:

IB, is open (union of balls P r ) , B "is open consisting of {+»}«{-»}

and (Sl»B = +.

Let us give a topology x on TO,* in such a manner that the closure

of HI in IB," is 1R» itself. To do this consider example 3 above, where

have taken the following function:

Here, i f r > n then P ^ = HL y {+-} u{-»} = & = P ^ and if r $ IT then

P ^ = {xtm.| arctg x > | - - r>w{+-} .

He shall denote by Pr the usual ta i l s of the Euclidean topology, i . e . give '

- l i -

r>

Hence none of the P r contains points of B while every P* and every P ^

contains points of 1ft. We observe that IH is an open set, in fact

B, = ^ > ^ P . To see this consider V x 6 Z (vhere Z = relative integers),

P2c(R = * \JP2cflL. For all x« IR, there exists n(ic) such that x« P2, ,,

vhere n(x) is the largest integer less than or equal to x.

Observe that all the open balls, those with real numbers as centre and

those with centre +» or -» have non-empty intersection with fiL Also it

is trivial to verify this for halls whose centres are reals, but we shall

verify this for P ^ or P1^ , i.e. we prove that ¥^n fR ?* $ for r > 0.

From the density of ft {the rationals p/q) in IH, 9

and also r > > — — > 0. Thenq p+q

such that r > •*• > 0

Let xtlR such that arctg x » jr - —— (such a point always exists) then

2 - arctg x • •—• < r = • x*^^,

The consequences of the above observations are that

1. B is not open but closed;

2. B is the boundary of IR.

Proof

1. We do not hope to find B from the balls P because thesethe x r r

have/empty intersection with B. Therefore, we examine P and P taking

their intersections

K-n KL * C ( r ' r t ) 9 real points

P^_n P^_ (» proper or improper subsets °f (B, Cr » r' « it

The argumtnt goes through for any intersection as far as it is finite. Hence

all points of B belong to the boundary of (R. K«* caniider the set B.

In fact, in every ball having a point of B as centre there exist points which do

not belong to B. Since the boundary of a set is closea, therefore B 1B closed.

2.. In every neighbourhood of +« t or of -"1 there are points of BL

and of B, therefore B la the boundary of m. It follows that tha closure

of Bt in Of 1 B E U B - B « , One sees immediately that the topology induced

by T on B is discrete: P rt B -j*+<>0» and Pr n B =\and Pr n B =\- " i

and Prn B - i

The topology induced on IB. is Euclidean because Prrt (R = Pr , the "oia"

Euclidean halls and

+»

R union of Euclidean balls if r > IT

aretgx>J-|.J u {+-}) n fc f i f f f ) =

ft Lf r=1Ti Lf

Every set of the type {x*IR[x > M) is a union of open balls

Proof

The topology T is Hausaorff.

We want to prove that V x ^ y«[R.# , 3 Pr and Pr> such thatPxrt Py " *• TJ»ere are three distinct cases

1. x i yfilR

2. x f y#B

3- x«iR, y«B .

Case 1

If x j y* IB then ]x-y| > 0 and

- I .ZCIR. x-z < r l

- l i t -

Putting r < ~ |x-y|, r ' < | |x-y| results in pJnP r ' = + . (in fact ve

know that 3 S e P ^ P ^ W U-d * !*-*! + |£-*l< f U-yl + - |x-y| < |x-y|which i s absurd.)

Case 2

Take x » +» and y = -» and r ,r ' < i to simplify the calculations.

Then

and the intersection

It is enough to pose r - r' J to attain P*\n P r = + . In fact one obtains

(•artcg x > 0the system I which does not admit solutions.

Larctg i < 0

Let xclR, y = +» and choose r' < it again to simplify the calculations.

Then

ft

w/Ken r'< arctg (x + r ) .

Also, i\ is easy to see that

arctg* > aretg (x-

< ircfcg (x + r)

Therefore the solution of the system of inequality

orctp 2 > arctp (x-r)

* < arttg (x + r)

* > ^ -r' > arctg (x+r)

belongs to p'rt PT_ . This system does not hare any solution sincearctg z > arctg (x + r) and arctg z < arctg (x + r) contradict each other.A Bimilar proof goes through for -» .

We would now like to give a topology on ft?, a topology T* whichconserves a l l the properties given .aWre except the closure of B. Let ustherefore define d* as given by the conditions (*l.2a) and (4.2b) and thatthe triangular inequality does not hold. Also define P+QQand Pr as

but we give a new definition of the open balls with real numbers as centre:

precisely V y frta, r « R.

if atcfc gy<r-]£

This tells us that the lialls with, real numbers as centre can be "richer" in

structure than those of Euclidean balls vtth th.e same centre and radius.

Let us nov define the open sets of the new topology as the arbitrary

unions and finite intersections of the balls Pr, P ^ and P ^ \jrelR. • One

gets immediately that (B.* Is open and also +.

«r

of If or -

Proof

The set B is open in the new topology: B = p'oP*

M

and d.*(oro°) <-rr 0<Tr)

Mere ast

and

Analogously, the induced topology on B is a discrete one.

Proof

The topological space

As usual we have three cases:

1- x i y«R2. x s* y«B3.

is Hausdorff.

Cases (2) and (3) are derived from. • the above discussion and hence wetake case ( l ) . It follows from Case (3) that there exists an (open) ball withcentre x which does not contain +» ( i t is enough that r < ^ - arctg x)and a ball with centre x that does not contain -» . The intersection ofthese balls is. another ball with centre x and i s Pr " {z«IP.| |z-x| < r} .Given P , the intersection of the two is empty when r + r ' < d*(x,y).

Lemma 3

(R, is open in the new topology T*.

Proof

V x^Dl, define r{x) * Minlj - arctg x, + arctg xj and r(x) > 0.

¥e shall then verify that

Since d»(x, +~) - ~ - aretg

therefore Dt:

£ r(x) and a«(x, — ) - j + aretg

One concludes that E? = R u B with (R open,

xtffiB open and (RMB = + . This means that BU* is not connected (see e.g.Dieudonne, Refs.8, 12.2 and 3.19). The topology induced by T" on 0V i sEuclidean, ip fact

ACKNOWLEDCMEHTS

One of the authors (K.T.S.) would like to thank Professor Abdus Salam,

the IntematlonBl Atomic Energy Agency and UNESCO for hospitality at the

International Centre for Theoretical Physics, Trieste.

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IC/79/2O J .T. HacI'iin.LaH and M.D. SCAnHONj Low-men-j rhotn- and >ls5tpop™li,flio«for- physical pions - Ii Mara identi ty an.'i c t i r a l treakm^ s t ruc ture .

J .T , MacKULLiir and M.D. SCABttONj Low-anerpy photo- and elactroproduotionfor physical pion3 - I I i Pbotoproduution phonomanology and extraction ofthe quark mass.

IC/79/22 B.K. PAL: Charm fragmentation function from neutrino data,INT.HEP.*IC/79/23 M.D. 5RISIV:\S: Collaosa postulate for ohsarvablas with continuous apecta.

IC/79/24 RIAZUBDIS and FAYYAZUDD1N: Gluon corractionH to non-leptonic hyperondecays.

IC/79/25 H.P. HITAL and U. NASAItfs Dialectronic racombination in sodium iao-IHT.REP.* electronic saquanae.

IC/79/26 MUBARAK AHMAD and H, SHAFT JALLU: Gel'fand and Taetlin techni v;a and1ST,REP,* heavy quarks,1C/79/27 Workshop on dr i f t waves in hi^h temperature plasaaa - 1-5 SeptemberIMT.REP.* 1978 (Reports and summaries).

IC/79/28 J.C.- E3T3VE and A.F. PACHECOi Renormalization group approach to theIBT.HEP.* ph»3d diafram of two-dimensional Hoisenbarf; spin systeoia.

IC/79/29 E. MAHDATI-HEZAVBHs Production of l i r h t leptonB in arbi t rary beamsIHT.REP.* via many-vector bo3on aichanpe.

IC/79/30 M. PAERINiLLO and M.P. T05I; Analytic solution ofthe mean sphericalIBT.REP,* approximation for a multicomponant plasma.IC/79/31 H. AKQAYs Thg leading ordar 'behaviour of the two-photon scat ter ingIffF.REF.* amplitudes in QGD.

IC/79/32 W.3. CHAIGIE and H.F. JONES: On the interface between aa»ll-p_ non-perturbativo and larpe-an^le perturbative phyaics in QCD and tBepaxton nodal.

IC/79/33 J- LOHEHC, J , PRZYSTAWA and A.P. CRACKllBLLj A comment on th» ohainINT.REP.* subduction c r i t e r ion ,

IC/79/34 M.A, KAHAZIE and D. STOREYi Supergymaiatric quantisation of linoarieadsuporgravity.

IC/79/36 J , TABSKI: Uamarka on a conformal-invariant theory of pravity.IHT.flEP.*IC/79/37 !•• TOTH: Additive quark model with s i r flavours.IHT.BEP.*IC/79/3B I . KONBOE and T. TSMESVARIi C lnulation of c r i t i c a l eiponents toINT.REP.* 0{l/n )•IG/79/39 A.O. BAHUTs Hadronio multiplets in terps of absolutely stableIHT.REP,* part icleas An already-unified theory.

IC/79/40 A.O. BAHUTj Stable par t i c les as 'building blocks of Batter .

IC/79/41 A. TAGLIACOZZO and E. TOSATTIi Effects of spin-orbit coupling onIHT.REP.* charge and spin density waves.

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IC/79/42 G.C CHIRARDI, C. OMERO and T. SfEBEH: 'juantum versus c lass ical laws forsequential decay processes,

IC/73/43 G.C. GHIHABD1, C. OHKHO, A. HIKI11I and T. WEBfiEt Small time behaviour ofquantum non-decay probability and Zeno'a paradox in quantum mechanics.

IC/79/44 HIAZUDDIK ant! FATTAZliDDINi (fcjark tnaas ratios due to ColdstoneINT.REP.* paeudoacalar mesons pair.

IC/79/45 FAYrAZUHDIN and HIAZTODIN: A model based on faufre symmetry group0 - 0 ^ I [311(3) I 3U(3)]o .

IC/79/46 W.S. CRAICIE and ABDUS SALAM; On the effect of scalar partons at abortdistances in unified theories with spontaneously broken colour symmetry,

IC/79/47 A.O. BARUTi Infinite-component wavtj equations describe r e l a t i v i s t i cIWT.REP.* composite systems.

IC/79/46 T.B. SKERRY: Comment on the question of faugo hierarchies.IHT.REP.*

ic/79/63

IC/79/69

lc/79/70

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ic /79/75INT.REP.*

S. MANOFFj Equations for ths pxavitationnl f ie ld and looal conservedquantities in tha general theory of r e l a t i v i t y .

V.P. FHOLOV and G,A. VILKOVISKYi Qiantum gravity removes c lass icals ingularit ies and shortens the l i f e of black holes .

M.A. RASHIDt A new expression for the Talni-MouMnoky harmonicosci l lator bracket.

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T.D. TOSOROVi Asymptotic numbers, asymptotic functions and distributions.

IC/79/51 A.O, BAfiTrT: Hapnatic resonanca3 between massive and mstsslesa spin-Jparticles with mafriotic moments.

^C/79/52 V. ELI AS: Gauce invariance and ferction maas dimenaione.

IC/79/53 HIAZUDBIS and PAYYAZUDDINt CL non-leptonio decays as a teat for KIUOHcorrections to non-leptonio hyperon decays.

IC/79/54 niAZUIDIH and PAYTAZUDDINi Non-leptonic radiative decays of hyperonsin a /-aur«-invariant theory.

IC/79/77 I.H. BL-SIRAFYi First and second fundamental "boundary value probl»»»IJPT.REP.* of Bpiral plate.

IO/79/7B k. SADItJ, M.A. KHAN and K.A. BHATTli Cluetering in correlated andIST.REP.* uncorrelated peroolative syatema.

Ic/79/flO A.O. BABOT, I . RABUPPO and a. VlTIELLOi On eleetrodynaBio* with i n -IHT.RKF.* ternal ferwionio exoitations.

IC/79/81 A.J. PHARGSi 3un rules over panarallied hyperpeoitotrio funotiont.

IC/79/56 B. ROBA3CHIK, 0. TROCER and E. WIECZOREK* Li<*t-con« eipansion of matrixIHT.flEP.* elements of current commutators.

IC/79/57 E. OAVA, R. JBNCO and C, OMEROi CP,,^ models and their non-abellanversion KPn_i« A study of tha quantum oroparties.

IC/79/83 I.H. EL-SI Ft AFYi Analysis of a boundary value problen of a circularINT.REP.* annular e laet io plate.

IC/79/B4 I.H. EL-SIHAFTi Boundary value problems of tbe linearised non-INT.REP.* boogeneoua Havier-Stokes equation* for tbe axlBymnetrioal slow notion.

IC/79/59 V.V. MOLOTKOV and I.T. TODOSOVi Frame dependence of world l ines forIWT.SEP.* direct ly intaractinr c lass ical r e l a t i v i s t i e part ic les .

IC/79/61 G. 3ENAT0RE, A. PARHIMELLO and '.'..V. T05I» Ontical absorption of diluteIHT.REP.* solutions of metala in molten s a l t s .

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IC/73/63 a . YU350UF; A p o s s i b l e r e a l i z a t i o n of jsinatein'a causal theory under-IHT.REP.* lyinff quantum mechanics.

IC/79/66 M.F. KOTKATA, E.A. HAHHOUD and :<.K. iL->:0U3LT: Kinet ic3 of c r y s t a lIOT.REP.* prowth in araornhous s o l i d and supercooled l iqu id TeSe2o uein^r LTA and

d , c . conduct iv i ty Eeaauremants.

IC/7J/B6

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IC/79/88

IC/79/89IST.REP.*

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10/79/96UTT.BEP."

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IC/79/98IMT.HEP.*

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0. CALUCCI, R. JEVOO, F. LEGOVIKI and K. PA VSR 1 P tad T violat ingelectromagnetic interaction of the quark in the instanton f i e l d ,w. MSCKLESTBURGj Aapeota of Sevan dim«naion»l r e l a t i v i t y .P. BUDINIi Quarks as conforatl aemi-spinors.

3 . FBRRARAi Formulation of auper^ravity without auperapace.

L.H. 3HEHATA» Boundaries of metaat&ble states in type II super-conductors.

A. KOOKERJES and S.C. ACARVALt Are aaiorthous Oe and SI frustratedepia elasaaa?3 , FSRRAHAt Spontaneous supersyometry bi-e&kinjr in Bupargravity.

V.B. OODWIN and S. T03ATTIi . Local field corrections to tbe bindingof aore excitons and shallow iuourltias in se>iaonduotora.

K.F. HOSTAFA, H.A. SBKAflY and M.A. AJIM£Di Magnetic e u e o e p t l b i l l t fi n v e s t i g a t i o n of some antiferroma*metic p'i + conolexea.

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IC/79/lCO r.A. AMINt The eich«nfe proDertv of iLodules.INT.REP,*

IC/79/107 a.Y.M. HASSAN and H.K.M. KANSO'JS: Halativigtie calculation of polarizedINT,HEP.* nuclear matter.IC/79/108 M.Y.M. HASSAN and 3 .5 . M0NTA3SEFi: On the thermal properties of nuclearINT,HEP.* matter with neutron excess.IC/79/109 J . S . WCOMAi Theory of absorption by exciton nolaritons in a spa t ia l lyIWT.REP." dispersive media.

IC/79/ l l l S, FERfiARA: Superapace aspects of supersymmetry and suporgravity.IHT.REP.*

IC/79/113 K.S. SINOWI and M,P. TO3I: Halation between "built compressibility andIHT.REP.* surface energy of alectron-hola l iqu ids .IC/79/H4 A.H. HAS3AH: Two-photon t rans i t ions to eioiton oolaritooa.INT.HEP.*

IC/79/115 G. AKBENIZ and A.O. BARUTs Cau^-invariant formulation of dyoniumINT,REP.* Hamiltonian on the sphere 3-^.IC/79/117 J . S . flKOMli Linear photon and two photon absorption by surfaceINT.HEP.* polar i tons .IO/79/ll6 A. VISINESCU and A. CORCIOVEIs Dechannelinj? in the HKB approximation.IST.REP.*IC/79/119 J*. APOSTOLs f in i te size effects on the plasma frequenoy in layered1ST .HEP.* slectron ^aa.IC/79/129 f .3 TOMCHEV and J .O. BHAHKOV: On the 3-d model for coexistence ofINT.REP.* ferromaj^netiam and auoerconductivity.IC/79/130 B.D. KATOILAHOV, K,T. PRIMATASOWA and V. IlETCHEVAi InterfaceINT.SEP.* in a class of beterojunctions "between diatomic semiconductors.