Introduction
For a non-empty set )( , let 1fJQ<)denote its power set.
Also let()(_, (:) and('{ ,
L1)
be two topological spaces· and
X
;;() ( [ ) be a family of subsets of
c: ~(?0rf
satisfying [
C
~ ('L)
1-: (X c::)~C'< c,) be such that for.
GE c, , 5-'(4)E ;zj(l:) then attempts have been in
a mapping
I
I
J
progress to investigate analytical properties of such a mapping
f . Clearly the
class of mappings with such assigned
property is larger ·than the class of all continuous mappings
: (;<.'c) --7 ('{ r,)
J
and depends upon the family
2J cc)
one chooses. While tracing the history backwards, one finds
that tak-ing
X
to be an Euclidean n-space
Euclidean topology (
and
(Y) L1)
En
with the
as the metric space (MJ
d.)
in 1961, C.Goffman and D.Waterman in [41] have studied the
properties of what is known as, an approximately continuous
cZl ( '[)
mapping referred to
as a
d~ns it y
topology in
En ·
Following this line of development, in 1965, A.P.Baisnab in
[1] has investigated
properties of mappings known as appro-
ximately continuous almost everywhere' by taking
be the space
E1
('{, 'L,):: E\
ex.) [)
to
of reals with usual topology and by taking
and by choosing
almost open subsets of
·E 1
~
(l:)to be the family of
- 2 -
In 1932, S.Kempisty in [43] studied for the first time
a class of mappings called the class of quasi-continuous
mappings over.an Euclidean n-space and investigated their
properties in the classical setting. Much later on in 1952,
W.W.Bledsoe introduced in [8] the notion of 'neighbourly
mapping'. These two notions have arisen independently and
have been reigning in.literature until 1958 when S.Marcus
in his article [44] showed that these two nqtions are identical. The works of W.W.Bledsoe in [8] and S.Marcus in [44]
however rest on the setting of metric spaces. The extension
of this theory over a general topological space has been made
possible by N.Levine in 1963 in his work entitled 'Semi-open
sets and semi-continuity in topological spaces' appearing in
[2]. The Definition of
s~mi-continuous mappings over a general
topological space is thus an extension of the Definition of
quasi-continuous mappings given by S.Kempisty in [43] or the
Definition of neighbourly mappings as given by W.W.Bledsoe
in
[8].
In [2] N.Levine has defined a set in a topological space
(X, t) to
A C [-cl(O)for some open set
0 E L where r- cl_ denotes L -closure. It is known that
(_-c-dc0) \ is nowhere ?ense in X . Consequent! y if A
is semi-open then A= 0 \j B where 0 E '( and B is a
be semi-open if
0
C
o)
nowhere dense subset with 0 ·n
property of a semi-open set
the definition is that if
~
~
B =¢ .
Another interesting
which follows readily from
is non-empty, thenso is
In:tA
- 3 -
L.
(Int denoting
-interlor). Further study on the algebra of
semi-open ·sets reveals that intersection of two semi-open
.
sets is not, in general semi-open, but union of an arbitrary
family of semi-open sets is semi-open. So the family of all
semi-open sets of a topological space
5. 0
~ [) denoted by
L) does not yield a topology on X . N.Levine
a mapping f : (X~
----7 ('( '[ ) semi'-continuous ( 5. c)
1
1
E. [ 1 implies .:f (Ci) t_ 5. 0 (X ~t). Since every
(X,
called
if
CX
Q
rJ
open set in ( )<.., ' [ ) is
-~emi-open, it follows that every
i : (X_. I) ~CY, r,)
continuous mapping
is also semi-conti-
nuous. An important
result of N.Levine in this connection is
;:f.oTm ct J:Je:t
The points of discontinuity of a S-~C.
mapping
.of the first
category in
(X,. L)
(y, "[1)
i_ f 11
if
J
Moreover if a sequence
('x . cD ~ (Y.)
where
(X., d)
dt)
and (
is a second countable space.
of
S .C .
mappings from
converges uniformly to
Y, d l)
are metric
5: (x, c\j-7(Y,~J
spac~s, then
f
is also S.C
Following this line of development we have shown (see
Theqrem 1. 3.~) that if a nett
ings;
(X
I
f
n £ D}
'1\ ',
r) ---j (Y d) converges
of S . C . mapp-
pqintwise to
I
then the set of points of discontinuity of .{
:f
E.
yX
is a set of
L) where (_ Y , &) is a metric space.
that if a net t :f
E.. DJ of S. C. mapp-
first category in ( )<.,
We have also found.
in?s from
j
:
11
'(\
(x ~ C:) ---7 ('( .~ 6l£.)converges
5 . C.
•
( )( , [ ) .
semi-uniformly (see
S; (X . t) ---1 (_Y d'l1.)
(Note that if a net of mappings {_ :f
n £. 1) J
(Y, "lJ..) converges uniformly to &E. y X,
Definition 1.3.2) ·to
is also
'1\ ••
5 t_ YX
then
1
'1\ ••
)>
j
- 4 -
then the net {
where ( '( ,
5n ~
ttl)
n E..l> Jalso converges semi-uniformly to 5
J
is a· uniform space).
We now move in another direction. In 1963, J.C.Kelly in
[13] observed that if one omits the requirement of symmetry
OXiOm ( i • e •
X (f.¢)
d. (X::_,~);;:. cl c~ 'X-))
I
d.
for a metriC
On a Set
X satisfying
cl(1-; -::J)::oif and only if (iff) ?C.:::.:i,; d..(X) 'l-)~d.(x:,.~-y)
(i.e. a distance mapping on
+ d.. c~' z) and d... (x' -:Y) ::: d..(_;j
a topology "[d.
I' 'J 0
J
»
iC.
J
then one still obtains
form a base for
lc;L : He
showed that
topology, but due to possible loss of symmetry
neither
li~
nor regular. In this connection he noted that
associated with such a non-symmetric metric
cl ,
quasi-metric
defined by
J [X-, 'Y) ~ "fJ_,
LtJ.. is a T1 axiom, Ld_ is
, where sets of the form '\_ .Y :
called a
there is another such quasi-metric
~*(-x..,~) = at(Y
the conjugate of
dL ,
cl
on
)(
):.x);
?tJ~f.X~o\.is
cl~
called
•
The lost symmetry has been restored by him not in a special
type of quasi-metric space, but in a bitopol6gical setting. So
we find generalisations of symmetric results such as Urysohn's
Lemma, Urysohn's Theorem, Tietze's Extension Theorem and Baire's
Category Theorem in
('f,
lQ., LeA*) under sui table separation
axioms which he defined in a general bitopological space. A
non-empty set
X
on
L and "\(b
(X ""[ ~)
which two topologies
defined is called a bi topological space
J
are
,
- 6 -
In [13] J.C.Kelly obtained the following result :
set
a
Let
(X, L 'f?l)
A
and a
"'[-
be pairwise normal. Then given a
1
L -closed
tl. S.C.
such that
and
set '8
\f.- J.... 5. C.
·£(A)=?
.and
An 'B
with
mapping
£ (B)::\
T0
Definitions of pairwise
,
=¢
S:
.>
--y-a -closed
there exists
X
L.O 1]
--7
1
·
pairwise
T 1 bi topological
(X, t:) the following result holds
spaces are now clear. As in
in a bitopological space.
Pairwise T.
~
·
~ Pairwise T 1
In a bi topological space
...:/ Pairwise
To •
'
(x T,'fl) ~ "[is
1
~aid
to be regular with
respectrdw.r.t.)-va, if for each ?C.E )< there exists a
base of V-closed sets.
(_X I"[) -v") is
L.-nbd
called pairwise regular
iff each is regular w.r.t. the other. The Definition of pairwise
completely regular space is framed similarly. In(X,l:,\(IJ)I [
is said to be completely regular w.r.t.
-ya
if for X£.
G E. [ ,
Lo., IJ
(X . L --ya)
there is a
"[ -u.s.c and\.(?' -l.s.c. mappingS : X~
such that
&(-x.).:: 0
and
J- (X\~)=\ . Now
is called pairwise cornplet~ly regular .
1
iff each is
completely regular w.r.t. the other. From the Definitions we
find that
Pairwise complete regularity _
?-
Pairwise regularity.
I.L.Reilly in (47] has dealt with separation properties in a
bitopological space.
- 5 -
The Definitions of·separation axioms in the setting of a
bi topological space
(X, L, -y'3)
are generalisations of their
analogues in. a topological space ( )<.) "() • Thus for example,
J.C.Kelly called a bitopological space ()(~1:J
T,_ if for distinct points
t- nbd.. (~-nbc!) U of X and
wise
')(.,
a
::J E.
-vo) to
)<.
be pair-
there exists a
-ya_ nbd.(t-nb&)Vof ~
U () V = ¢ . Now if (X~"'[) -yo-) is pairwise
Hausdorff, then both L and "\((1 are T topologies. J. C.
1
such that
Kelly in [13] showed that if "[ and
-yo
are induced by conju-
gate quasi-metrics on )( then the converse also holds i.e.
(_X~ r ~ -ya)
is pairwise
T,_ .
(X ld
Results in
.I
J
lq ~)
reveal that quasi-metrics are related to real-valued semicontinuous mappings in much the same way as metrics are related
to real-valued continuous mappings as the following result by
J .C.Kelly illustrates :, For a fixed X.. in X
Ld_-
and '[cl~-
'U.. · S. C
for a fixed
cl ( ?(, "::J)
in )<.. .~
'.Y
Lc;\ ~ - U.. S.C.
mapping of
A bi topological space
( . .5. C:
X
set
-closed and
U
and that
and a
B
L. -open
::J ;
is a (cl _ { . S . C .
And
and
.
(X, '"( )-ya) is
'
L
c:l(x;y)is a
mapping of
mal if for each pair of disjoint subsets
is
J
called pairwise nor-
A B C)<:
~
where
is \ID-closed there exists a \(0 -open
set
V
such that
AC U and B C V
- 7 -
Covering axioms. in a bitopological space are defined such
that they generalise their analogues_ in a topological space (X
J
t).
Ih this connection we state three Definitions of compactness
which are in use in a bitopological space. A bitopological
(X, [, -ya) is said to be compact if every open cover
'1.1 C [ u -ya admits of a finite subcover. A bi topologica 1
space ('I-- L ) \.(3)
is said to be pairwise compact if each
"[ ( -va)-closed subset C .:f X is -ya{'t)-compact. We have
space
J
called such a space as bi-pairwise compact in order to distinguish it from next that goes by the same name. A bitopological
space is called pairwise compact if each pairwise open cover~
(i.e.
6Le
containing at least one non-empty member of
at least one non-empty member _of
-ya )
finite subcover. Now we remark that if
then the space
C.
L
V
lfO
ex . L I~)
(:
and
admits of a
is compact'
X is L. -compact and also \(0 -compact. Clear! y
a bitopological compact space is pairwise compact. Again if
(X , L \.(a)
J
(X, L) is
is hi-pairwise compact, then '[:;. \(0
says that
compact if every proper closed subset of )<. is
compact, but reverse is the case in a topological space wherein
we get, if
(X,C::)
(including
~)
is compact, then every closed subset of
X
is compact. Thus this Definition fails to gene-
ralise that of a topological space ( "~--.
J
L) . The
Definition
of pairwise compactness is more appropriate in the sense that
every cover is linked with both topologies. So we have made
much use of this Definition. The following results run parallel
to their counterparts in a topological space
(X, L)
:
- 8 -
T :2-
A pairwise
(X~ L
and pairwise compact
1
-ya) is
pairwise normal. Also a pairwise regular and pairwise compact
(X L ,-yoJ
is pairwise normal. Weakening pairwise compactness
I
to Pairwise Lindeloff (i.e. each pairwise open cover has a countable subcover) we have proved in [38] among other results that
each pairwise regular and pairwise Lindeloff space
(X L -ya)
I
1
is pairwise normal. Also in [38] we have shown that product of
pairwise completely regular bitopological spaces is pairwise
completely regular. Some of the above results can be found in
[19]. We have filled some gaps to that extent only as to create
a situation for us based on which we can carry on further work.
It is now proper to state two important results of W.J.Pervin.
For this we first recall the following Definition of W.J.Pervin
from [17] :
A family
1£.
of subsets of
XxX
satisfying the follow-
ing axioms :
U C.. 61.t
(U .1)
Each
(U .2)
U, Vt.6f1_
(U.3)
V E 6(1
contains the diagonal
implies
V('\ V E.
(U.4)
implies
Any
V
~
implies there exists a
'VoVc:U
C.
)
V £_ ~
such that
)
X:><.X satisfying U C. V
V E. ~·is
)
~(_X) ;
where
U E OJ}_
said to define a quasi-uniformity on
X .
The quasi-uniformity becomes a uniformity lf it satisfies 1 in
addition the symmetry a·D(iom :
- 9 -
0U.
(U. 5)
UE
where
U~-tC~/Y):
Taking
i__
u
implies
UE_ 4.2_
(X) :
there is a unique
J
u}.
J
as the family of all )!lbds of X. ,
L'U.
topology
Conversely, every topological space
induced by
CX, [)
6]1
X .
on
has a compatible
quasi-uniformity ~ consisting of all sets which contain
finite intersections of sets of the form
(G
xG) u (Cx\G)xX)
Et~
where
rt is now easy to verify that the family
Of.i*_ i U~ UE6!fJ
also defines a quasi-uniformity, called the conjugate of
being denoted by ~~~
the induced topology on
)(
given topological space
(X,"[) ,
•
6],1 ,
For
W.J .Pervin in [17] has been
able to lay out a scheme for construction of a quasi-uniformity
ot£. p
compatible with [
• We also find in [50] that authors
R.Nielsen and C.Sloyer have given a construction for a quasiuniformity
£111. N
compatible with
'[
• It transpires that
<fLe_ p C 6)_Q N
This inclusion relation rises to equality through the work of
W.Hunsakar and W.Lindgren in [49] wherein we find that
6le_ p
= 6liN .
It is known that a topological space
iff it is completely regular. But
(X, L) is
uniformizable
Cx , [ tStt) is not complete! y
- 10 -
~~
regular i f the topology
~
is induced by a quasi-uniformity
• However the situation has been regained in the work of
E.P.Lane in [14] in a bitopological setting. In this connection
we no'f;Jstate the result of E.P.Lane
A bi topological space (
Y... , L
J
"V'Z') is
pairwise completely
regular iff it is quasi-uniformizable (i.e. there exists a quasiuniformity
6li
t ::.
such that
r
d(l
and
-yo:::. [
'1.1_
* ).
About 1963 W.J.Pervin in [18] generalised (Efremovich) proximity by dispensing with symmetry axiom of proximity and obtained what he called a quasi-proximity. His Definition runs as
follows
of
X
A binary relation
0
defined on the power set
is called a quasi-proximity on
(~(X))
)( if it satisfies the
following axioms :
(P.l)
Aa(BuC)
iff
A oc
iff
AoB
B o C.
or
A oC
or
and(AvB)oC
.)
Bt¢
(P.2)
A0 B
(P.3)
AnB:f: ~
implies
AoB
(P.4)
(Strong ·axiom)
AjB
implies that there exists a subset
e-c.X
such that
If, in addition,
(P.5)
A0 B
proximity on )(
space
A:f-¢
implies
0
and
A 1C
'
J
J
and
X\C ~B
satisfies the symmetry a~iom :
implies
B 0
A~ then C)
is called a (Efremovic)
• In [18], it was shown that every topological
(:><.., L) gives
rise to. a quasi-proxirni ty space
by defining the quasi-proximity
(J
as follows :
{_X,
0)
- 11 -
A 'd B
iff
A n [- cl (B) =f:
¢,
(X ~0)
Conversely, every quasi-proximity space
becomes a
topological space, if the closure operator is defined by
cl (.A) ::: t-x. : t X.} 0 A J for
A of X . Thus
any subset
he gave an alternative approach of defining a topology. Also
he showed that symmetry axiom (P.5) is equivalent to the complete regularity of the topology ~O
is completely regular, but
.
Thus a proximity space
(X La) is
J
not so when
a
is a
X .
quasi-proximity on
Guided by E.P.Lane's work in [14] we have defined in [37]
(S on a set X(t~J(see Definition
Associated with ~ there is another such ~*given by
a quasi-proximity
A ~* B
of
~
iff
CX\ B)~ (X\A) called
• The topologies induced by
pectively denoted by
7
L~
and
@ · and
rL<e~
.
results we have shown that if a bitopological
is pairwise normal with.both topologies
TI ,
3.2.1).
the conjugate
~
@..
are res-
In [37] among other
space~)(~ ~J ~)
then there exists
a compatible quasi-proximity given by
A~ B
iff
-ya_ c.Q.
CA) n [-J (x\ B)=¢ ·
Every pairwise completely regular space is not pairwise normal
(see Example 2.2.2). However it follows that a pairwise completely regular bi topological space (X,"'()
-yo) is
mal if there exists a compatible quasi-proximity
A~ B
iff
pairwise nor~
given by
-ya_c_t(_A)n r-clCx\B)=¢·
- 12 -
While searching for a unique quasi-proximity we have found in
[39] a special type of pairwise normal space in connection of
which among other results we have shown that pairwise equinormal spaces admit of a unique compatible quasi-proximity. Here
it is worth remarking that W.J.Pervin's quasi-proximity
~
and our quasi-proximity
on a set
0
)( are equivalent
(see Theorem 3.4.1). W.J.Pervin in [16] defined a mapping
cX
I }-va) ---7 (Y ~ r, .'1} to be continuous iff :5 ', (_x.' L)
-7l'Y, t 1)is continuous and j : (X J-yo)
) (Y ~)
f ·.
J
J
is continuous.
in quasi-proximity setting it is to be
H~wever
:f : (X a) ---? ( y 0 I )
then j ·. (X o., <1*) _....,) (Y , o ;o,~) need
1
noted that if
J
J
J
where
o"*
gate of
is the conjugate of
0\
0
is continuous'
not be continuous
0 *1
and
is the congu-
(see Example 4.3.1). The situation looks nicer
in respect of quasi-proximal mapping, defined to be a mapping
which preserves the proximity of sets. i.e.
is called quasi-proximal iff
l j oA
Since X
implies
A aB
l f [x.) J 0\ f
f:
implies
(A)
(x . . o)---?(Y . . a,J
j- {_A)
a, j CB).
we have
j (La-d (A)) c C0 ,-d. ( fCA))
And this implieS
X.' L oJ -----'!> (_y' La,) is continuous • I t is
now clear from above result that if .! : C x " 0) ~ (Y) a,)
is quasi-proximal, then f : (X o)t)
( y ) 0 ~)
s:c
J
is also quasi-proximal and then
j- : (x, t
0
,
t0 ~)
~ (Y, La, ,La:~)
- 13 -
is continuous. The converse also holds if the domain space is
'
pqirwise compact. Also we have established that every continu-
:f: ("i-- _, o., oj~ (R) P, P~
ous mapping
is pairwise equinormal and
..P
(where
(x,a~ o*_)
is the quasi-proximity to induce
the upper topology on the space R
of reals) is a quasi-proxi-
mal mapping.
Let (
mappings
Dt1)
Y_,
be a quasi-uniform space. Then a net of
Lrn : . n €. D J
from
:f
verge quasi-uniformly to
E.
X
yX
y
to
is said to con-
iff for
U E. 1Q.. .J'
eventually for all
:f n Cx-)
f_ \j
Similarly, let
t :f
't'l '
Then
Y)
* lf [?tJ J
(Y _, 0)
J
E. D
l fn J
and
eventually for all
be a quasi-proximity space and let
be a net of mappings from
)<.. to '{ .
5 E.. Y X
A c X and B c y, 5cA) ~ B
implies
eventu~lly and S (A) '¢* B
is said to converge quasi-proximally to
iff for any two subsets
Jn (A)~ B
f '1"\ (A) cp'* B eventually.
implies
~
Quasi-uniform convergence
~
We have shown that
Quasi-proximal convergence
Pointwise convergence.
Regarding limiting behaviour of a net of quasi-proximal mappings:
c)(./ 0)
5- E. yx,
Also if a net
~ (Y
0 I)
J
converging quasi-proximally to
we have proved that
1_
Jn
hi topological space
: Y'\
l
€.. Dj
of continuous mappings from a
X / [ , -yo)
converges quasi-proximally to ~
is also quasi-proximal.
to (_
, then
f
Y, 0 _, 0 *)
is continuous.
- 14 -
Finally we have compared the notion of quasi-proximal convergence with the notion of continuous convergence (see Definition
4.4.1 and Theorems thereafter).
Another~part
of our work is concerned with the problem of
obtaining fixed point for mappings on generalised spaces. Initial
difficulty seems to centre around for introducing notion of completeness in the space and for introducing notion of contraction
for a mapping acting on a generalised space. Thus we see that
theory of fixed point has not developed in a uniform space as
fast as in a __ metric space. First attempttowards defining completeness in a bi-quasi-pseudometric space
(X ~ d ) d*)
appears
in J.C.Kelly's work ~n [13] wherein he used d-cauchy sequences
in the space. Main obstacle herein is that a d-convergent sequence need not bed-cauchy (see an example in support in [13]).
Also work of R.Stoltenberg in [48] calls for our attention in
this respect. There author has found difficulty in extending
definition of a cauchy net in a quasi-uniform space
(X~ 611.)
and concluded that no completely satisfactory definition could
be found. On the other hand bringing in contracting behaviour
of a mapping in a setting beyond the realm of uniform spaces
seems to be another hurdle in an attempt to prove contraction
Principle type Theorem in a generalised space. In this respect
we follow J.Dugundji's technique. J.Dugundji in [26] introduced
the notion of positive definite mapping on a metric space and
obtained a fixed point theorem therein. The notion of positive
definiteness was extended to a uniform space by K.K.Tan and
C.S.Wong in [27] and by C.S.Wong in [28]. In this chain,
- 14
(a.)
A.Chaudhury and A.C.Babu in [29] have further extended the
result in a quasi-uniform space. In our work our space will be
a quasi-proximity space. Following S.LeaJer in [30] we have
defined a gauge on a quasi-proximity space
that the family of all gauges on X
mity
(J
(X_,
o)
such
generates the quasi-proxi-
on it. With the aid of gauge in a quasi-proximity
space we have been able to reach a notion of completeness of
the quasi-proximity space. And through extension of notion of
positive definiteness of mapping in the space we have arrived
at a fixed point Theorem for such a mapping.
The final chapter openswith an aim to establish a prototype
of Banach Contraction Principle Theorem in a separated quasiproximi tv space
(x .~ a)
where
L"a
is a Hausdorff topology.
Applying W.J.Pervin's result that every topological space is
quasi-proximiza~le,
our result atonce extends to a Hausdorff
topological space. The chapter closes by proving some results
on fixed point theorems for a set valued mapping in a metric
space.
The thesis is divided into six chapters
In Chapter 1, we consider semi-open sets and semi-continuous
-'
mappings on a topological space. With the aid of semi-uniform
convergence of a net of
mapp~ngs
it has been proved that the
limit mapping of a net of semi-continuous mappings under semiuniform convergence preserves semi-continuity. Nature of points
of discontinuity of a semi-continuous mapping has been investi-
- 15 -
gated. In the sequel, a generalisation of a theorem of A.Huber
[6) has been obtained. The chapter ends with the study of the
boundary behaviour of a real-valued semi-continuous mapping
over the upper half-plane of the Euclidean plane
E~
with
values in a metric space. Our findings then encompass those of
A.M.Bruckner and C.Goffman in [7].
Chapter 2 entirely deals with general bitopological spaces.
In Chapter 3, among other results we have established here
analogue of Urysohn's Lemma in a quasi-proximity space. The
equivalence of W.J.Pervin's notion of quasi-proximity and the
quesi-proximity defined by us has been shown. Characterisation
of pairwise completely regular bitopological space has been given
in terms of quasi-proximity. Searches have been made for bitopological spaces admitting of a unique quasi-proximity compatible
with its topologies.
Chapter 4, concerns with
quasi~proximal
mappings. Limiting beha-
viour of such mappings have been studied in details in respect
of different kinds of convergence.
Chapter 5, for the first time contains our findings on fixed
point Theory in a quasi-proximity space. Introducing notion of
positive definiteness of a mapping and using the aid of gauges
to generate a quasi-proximity we have proved a fixed point
Theorem for such a mapping in a quasi-proximity space.
- 16 -
We have continued our investigation on fixed point theory
in the last chapter i.e. Chapter 6. It starts to introduce the
notion of a contraction of a mapping on a quasi-proximity space.
A contraction-principle type fixed point theorem has been established. A weak contraction in a separated, complete uniform
space is defined and a fixed point theorem has been obtained
therein. The final section of this chapter contains results on
fixed points for set-valued mappings acting on a metric space.
While presenting the work contained in this Thesis a number
of books and journals have been consulted. The references to
these books and journals have been appended at the end of Thesis,
and the numerals corresponding to the references in the bibliography have been inserted at the places where they occur.