FUNCTIONS BASED ON COMPACTNESS

Kyungpook Ma.t.hemat.ica.1 Journa.l

Volωne 31, Number 2, Decembcr, 1991

FUNCTIONS BASED ON COMPACTNESS

M. E. Abd El-Monsef

,

R. A. Mahmoud and A. A. Nasef

Compactness is one of the fundamental topological properties on spaces‘

So many types of functions are dcfìned before depending on corresp~)!1ding

types of compact scts. This leads us to devotc this paper to construct a new c1ass of functions depending on the notion of RS-compactness due to IIong calling it RS-continuous functions. Some characterizations and several properties of this new type of functions are investigated. Rela tionships between RS-continuity with other corresponding ones aod its effect on somc known topological spaccs are studied. 10 the last sectioo we generalize all knowo types of functions which based on compactness under the (-continuous concept.

1.

Introduction

Scveral types of non-continuous functions which are related to different typcs of compact and c10sed subsets have been established and studied by a great staff of topologists. Firstly

,

in 1970

,

the concept of C-continuity has been defìned by Gentry and Hoyle [1]

,

one ycar latcr Park [2] gave othcr typc callcd C나ontinuous functions. 1n 1975

,

Long and IIamleU [3]

have introduced and studied thc notion of H-continuity. Also

,

Kohli [4]

in 1981

,

invcstigated the c1ass of L-continuous functions and in thc samc year Long and lIerrington [5] have obtained the para-continuity notion 1n 1984

,

Malghan and IIanchinamani [6] have introduccd the concept of N-continuous functions

,

also in 1986

,

by an independent study Abd EI- Y10nsef and Mahmoud [7] defincd and studied the same type uncler the

Received December 4, 1989 Revised November 5, 1990

Key words and phrases: C'-continuous, C'.-continuous, H-continuous, N-continuous,

N C-continuous, para-continuous, L-continllous, S-continllous, s-continuous, κ5-

cornpact subset relative to its space, RS-contínuous, Ç'-con^iÎnuolls 1980 AMS slIbject classification code. 54C10

275

276 M. E. Abd EI-Monsef, R. A. Mahmoud and A. A. Nasef

concept of NC-continuity. In 1984, Di-Maio [8] gave only the notion of S-continuous functio 며 but in the same year Lashien [9] ^introduced the same class of functions and investigated some of its characterizations and studied several of its topological properties. Recently

,

the authors [10] constructed the concept of s-continuous functions and many of its properties are established.

Generalization of functions is one of the interesting subjects for some topologists. e.g. In 1981

,

Ha때ett and Herrington [11] introduced the con cept of]'ιcontinuity which generalizes C-continuity

,

C*-continuity and H- continuity. Also

,

in 1986

,

Abd EI-Monsef

,

and Mahmoud [7] investigated the class of a* -continuity which generalizes C-continuity, H -continuity and NC-continuity

In this paper

,

we introduce a new class of functions called RS-continuity this new concept depending on RS-compact subsets in the scnsc of Noiri [12]. This new type is independent from C-continuity but it is wcaker than s-continui ty

,

so many res비 ts in [10] are improved. Some characterizations of RS-continuity and many of its properties are obtained. Results of its cf fects on known topological spaιes are investigated. Also, the last section of this paper is devoted to define the concept of ç-continuity as a generaliza tion of several types of functions based on corresponding compact sets like C-continuity

,

C*-continuity

,

H-continuity

,

N-continuity

,

S-continuity

,

s continuity

,

RS-continuity

,

paracontinuity and L-continuity. Finally

,

we investigate some of its characterizations and fundamental properties

2. Preliminaries

Throughout the present paper X, Y and Z always denote topolgical spaces on which no separation axioms are assumed unless explicitly stated The closure (resp. interior) of any subset W of X , will be denoted by c1(W) (resp. int(W)). For a space X, recall that W C X is regular-open if W = int(c1(W)) and RO(X) denote the collection of all regular-open subsets of X. Also, W

c

^{X is said} to be regular semi-open (Cameron,

1979) if there exists H E RO(X) such that H C W C c1(1I) and wc denote by RSO(X) the collection of all regular semi-open subsets of X.

Definition 2.1 [12]. W C X is called RS-compact subset relative to X

,

if for every regular semi-이)en cover {UQ a E \7} of W there exists a finite subfamily \70 ^{of \7} such that W C U{intUQ : a E \70}

Definition 2.2. A function

J

X • Y is said to be C-continuous

Functions ßased on Compactness 277

[1] (resp. C*-continuous [2], H-continuous [3]

,

L-continuous [4], para- continuous [5]

,

N-[6] or NC-continuous [7]

,

S-∞ntinuous ， [8

,

9]

,

s-continuous

,

[10]) if for eacb x E X and each V E r(Y) containing J(x) having a com- pact (resp countably compact

,

H-closed

,

Lindelof

,

paracompact

,

N-closed

,

S-closed

,

s-closed) complement

,

there exists an open set U C X containing x such that J(U) c V.

Definition 2.3.

J :

X • Y is called

(i) weakly-continuous [13] (resp. B-continuous [14]) if for each point x E X and each open neighbourhood V of J(x)

,

there exists an open neighbourhood U of x such that J(U) c c/(V) (resp. J(C/(U)) C.c/(V)).

(ii) almost-open (resp. weakly-open) [15] if J(U) c int(c/(J(U))) (resp. J(U) c int(J(c/(U))) for every open set U of X

Remark 2.4 [12]. RS-compactness and compactness are independent of each other even though the space is Hausdörff

Recall that a space X is extremely disconnected (E.D.) if the closure of each open set is open

Proposition 2.5 [12]. Lel X be E.D. Then Jor a subsel W oJ X

,

the followings are equivalent:

(i) W is RS-compact relative to X (ii) W is S-c/osed relative 10 X (iii) W is N -c/osed relative to X.

(iv) W is quasi H -c/osed relalive to X.

Proposition 2.6 [16]. Let f : X • Y be a Junction

,

then C -continuily

,

N -continuity and continuity are equivalent μ Y is compact

Proposition 2.7. Let f X • Y be para-continuous [51 (resp L- continuous μ}) such that J(X) is conlained in a closed paracompact (resp Lindelöf) subsel ofY. Then f is continuous

Propostion 2.8[12]. RS-cor때actness relalive 10 its space is preserved undel. one of the following

(1) The function is almosl-open and weaklν-continuous (2) The function is weakly-open and B-continuous.

3.

On RS-compact

Subsets Relative to its Spaces In this section

,

we obtain a new characterization of RS-compactness by using the concept of ß-openess (Abd EI-Monsef, et al. 1983) which is

278 M. E. Abd EI-Monsef. R. A. Mahmoud and A. A. Nasef

defìned as: W

c

^{X is ß open iπ W}

c

^{c/(int(c/(W))).} Some properties of RS-compact subsets relative to its spaces will be investigatαl

Lemma 3.1.

[1 A;,

^{i = 1}

,

^{2 are RS} -compact sets l'elalive 10 a space X

,

lhen A

,

^{U A2 is RS-compact relative to X}

Proof Let {V_{o :} ^Q E \7} be a regular semi-open cover of A1 U A2' then it is a regular semi-open cover o[ A;

,

^{[or i 1}

,

2. Since

A;

is RS-compact relative to X

,

then

,

there exists a fìnite suþ[amily \7‘。 of \7 such that

,

A;

c

U{int(Vo) ^{’ Q} E \7;0}, i = 1

,

^2. Therefore

,

A₁ U A₂ C {int(Vo ) : ^Q E

\710 ^{U \72}

01.

^{So, A}1 U A2 is RS-compact relative to X

Corollary 3.2. The intel'section 01 two open sets having RS-compact complements is also open having RS -compact complement

P^7'00f Let U and V be two open sets o[ X having RS-compact com- plements

,

^so

,

^U n V is open

,

^{and also X \(unv)=(x \}U)U(X \ V) is RS-compact

,

by Lemma (3^,1).

T. Noiri [17J proved that any si떼eton o[ a space X is N -c1osed, ancl the present authors [1띠페100이J stre인eng망thened this result and showcd tμhat an singleton o[ any spac않:e 1S S순-c1 osed. Other improvement will be obtaincd theroughout Lemma (3.4) a[ter establishing the follwing proposition Proposition 3.3. A subset W 01 X is RS -compact relative 10 X i

.tJ

^eve'Jν

ß-open cover 01 W admils a ηnite sublamily

,

the interiol's 01 lhe c/os^U1'es 01 whose members covel'S IV.

P^7'00f Necessity

,

Lct {U; ^‘ i E

I}

be a β-open cover of W

,

then {c/(U

,)

i E

I}

is a regular c10sed cover of W. Since W is RS-compa.ct rela.tive to X

,

then there exists a fìnite subfamily 10 o[ 1 such that W C U{inlcl(U;)

i E lo}

Su태 ciency. Let {U; i E

I}

be a regular c10sed covcr of W

,

then it is ß-open cover. From hypothesis

,

there exists a. fìnite subset 10 of I such that W C U{ intc/( ι) : i E lo} = U{intU; : i E Io}. IIence W is RS-compact rela.tive to χ

Lemma 3.4. A singleton 01 any space is RS-compact

Proof Let X be any 5pa.ce

,

x E X

,

a.nd {A; i E

I}

be a. ß-opell cover of {x} in X . Then {x} C U{ Ai i E

I}.

I-Ience

,

[or some i

,

x E A; C intcl(A;)

,

a.nd 50

,

{x} C U{intcl(A‘) : i E

I}.

^{Therefore {x} i5}

RS-compa.ct

Lemma 3.5. Let A E RO(X) and B E RSO(A)

,

then B E RSO(X ).

Functions 8ased on Compactness 279

Proof Since B E RSO(A)

,

then there exists U E RO(A) such that U

c

B C clA(U) C clx(U). By regula1'-openness of A

,

we have U E RO(X) and hence B E RSO(X).

Theorem 3.6. Let X o ^E RO(X) and A

c

X o

c

X. Then A is RS- compact relative to X o if A is RS -compact relalive to X

P^1'00 f Let {V^{a ’} _Va E RSO(Xo)

,

^Q E \7} be a cover of A

,

^{then Lemma}

(3.5) shows that _Va E RSO(X) [01' each ^Q E \7. Hence _{Va : ^Q E \7, _Va E RSO(X)} is a cover o[ A whose elements are RS-compact relative to X

,

50 there exists a lìnitc \70 o[ \7 such that A C U{intx(1ι) : ^Q E \70}, Sincc X o is regular-open in X

,

then intX(Va) = inlxo(Va ) foreach ^Q E \70. This implies that A is RS-compact relative to Xo‘

4 . RS- continuou s fun ctions

Definition 4.1. A function f X • Y is callcd RS-continuous

,

if for each x E X and each open set V C Y containing f(x) having RS- compact complemcnt

,

therc exists an open set U C X containing x such

thλt f(U)

c V

Some characterizations of RS-conlinuity are establishcd throughou t thc following theorem

Theorem 4.2. Fo^l' a f^1lnclion f : X -• Y

,

the following statements are

eq 1l ivalen나t . .

(a매) f i염s RS -contin때l

(b비) The inve

,

'se image of each opeη s1lbset of Y having RS -compacl complement is open in X.

(c) 1f F is c/osed and RS-compacl s^1lbset of Y

,

lhen f-' (F) is closed in

X

(d) For each x E X and each πet {Xa}aE<;7 삐ich conve'yes to x

,

the net

U

(x a )} ^aE<;7 is event띠lIy in each 01Jen set co，띠ining f(x) and having an RS-compact complement.

Proof (a) • (b) : Let V E r(Y) and Y \ V be RS-compact, then f01 each x E f-'(V) thcre exists an open set U of X containing x such that

f(U)

c

^{V‘ Thus}

,

x E U C f-l(V) which mcans that x is an interior point of f-l(V), therefore f-'(V) is open^‘

(b) • (c) : Let

F c Y

^{be a closed and} RS-compact subset o[

Y ,

then Y\F is open having an RS-compact complement

,

by (b)

,

f-'(Y \ F)=

X\f-' (F) is open. IIence f-l (F) is closed in X

280 M. E. Abd EI-Monsef, R. A. Mahmoud and A. A. Nasef

( c) • (a) Let x E X and V C Y be an open subset containing f(x)

,

and Y \ V is RS-compact

,

then x

if.

J-l(y\11) = X \ J-I(V) which is closed. Therefore J-I(V) is open containing x. Put U

=

J-I(1I), then f(U)

c

V.

(b) • (d) Let

{X"}"E"

^{be a net in}

^X

^{which converges to x and}

let V E r(Y) containing J(x) such that (Y \ V) is RS-compact. Then x E f-l(V) E r(X) and therefore {x"}"E" is eventually in J-I(V). Hcnce

{J(X")}"E" is eventually in V

(d) • (b) Let V E r(Y) having an RS-compact complemcnt. To show that J- I(V) E r(X), consider the converse. i.e. If x E J-I(V) such that f-l(V)

if.

N(x) { where N(x) is a neighbourhood of x}. Then

,

thcre is a net {x"}"E" in X which converges to x and rrllsses J- I(V) frequently^‘ This shows that the net {J(X")}"E" misses V frequently

,

which leads to contradiction

Remal.k 4.3. lmplications in Diagram (1) give the connection betwccn RS-continuity and other corresponding types.

Continuity • Para-continuity [5J

/

H - continuity [3J • N[6JorNC - continuity [7J

\ •

S - continuity[8,9J • s - continuity [1 이 Diagram (1)

• L - continuity[4J

l

• C - cont.[

lJ

^{• C'cont.[}

2J

z ’

• RS - continuity

lt is evident that RS-continuity and C-continuity are indepent conccpts { See Remark (2.4) }. This shows that previous implications arc not

,

in general

,

reversible.

Theorem 4.4. Let J : X • Y be a Junclion and Y be cOlnpact

,

E. D Then the Jollowing types oJ Junctions are equivalent

(i) J is RS-continuous.

(ii) f is N -continuous.

(iii) J is S-continuous.

(iv) f is C-continuous.

(v) f is H -continuous.

(vi) f is continuous

P1'00f F'ollows directly by using Propositions (2.5) and (2.6)

Theorem 4.5. Let J be a Junclion J1'Om a space X into a αmpacl and E.D. space Y

,

then RS -conlinuity aπd pamcontinuity (resp. L. continu-

Functions 8ased on Compactness 281

ily) are equivalenl μ f(X) is conlained in some c/osed paracompacl (resp.

Lindelöf) subset of a compacl space Y.

Proof lmmediately by Theorem (4씨 and Proposition (2.7)

In what follows

,

we discuss another characterizations of RS-continuity and the relations between it and other corresponding ones by considera- tion of a special topology on Y. For a space (Y,(7), if we take all open sets having RS-compact complements a.s a ba.s

e

^{for a new topology} ã on Y. Likewise

,

the collection of all open sets having compact (resp. H- closed, N-closed, S-closed, s-closed) complements, as a ba.se to generate topologies (7' [18J (resp. (7' [3J, 히7]， (75 [9], δ[1 이) on Y.

Rema7.k 4. 6. One can notice that (i) (7' C (j C (7' C (7.

(ii) ã C â C (j C (7' C (7 (iii) δ C δ ζ (75 C (7' C (7

Rem^a7.k 4.7. lt is clear that spaces, (Y，(7)， (Y，(7')， (Y， δ) ， (Y, (75), (Y， δ) ，

and

(Y ,

δ) represent a complete distributive lattice

,

^{a.s shown in Fig (3)}

단&쇠」

‘

r /

/

(Y. <T i

E웅브l

From Fig (2)

,

we establish the following results.

‘

^Y.CI')

\Y. :l'SI CY.ãJ

0',;'

Th븐orem

4.8. A funclion f (resp. J',f''y,f5,

Î)

is RS-conlinuous iff f is continuous (resp. C -conlinuous, H -conlinuous, N -conlinuous,

S-coηtinuous， s-conlinuous), 뼈enever ^I (Y,(7) • (κ δ ) (resp. i

282 M. E. Abd EI-Monsef. R. A. Mahmoud and A. A. Nasef

(Y

,

a') • (Y， δ) ， i : (Y

,

a*) • (Y， δ )， i : (Y

,

õ') • (Y， δ) ， i ^‘ (Y

,

as) • (Y

,

δ)， i : (Y

,

8) • (κ ã)) is cOlltilluous (resp. C-coπtilluous， H -colltilluous

,

N-continuous

,

S-continuous

,

s-continuous) and i-¹ is RS-contilluous Proof We prove this theorem in the first case while others are sirnilarily

Let

f

^’ X • (Y, a) be RS-continuous and Ict V be a basic open set in (Y

,

δ) ， then V is opcn in (Y

,

a) and has an RS-연mpact complement By Th∞ rcmJ4.2) ， we have f-l(V) E T(X). Hcncc f is continuous. Con- versely

,

let

f

X • (Y

,

a) be continuous and let V be open in (Y

,

a) having an RS-cor매act complcment. Then V is open in (Y， δ) ， 50 that f-l(V) is open. By Theorem (4.2) f: X • (Y

,

a) is RS-continuo^1l8. Wc also notc that the identity function i : (Y

,

a) • (Y

,

δ) is continuous

,

and i-¹ is RS-continllo1l8.

Theorem 4.9. Let f be RS -collti^1l1

‘

^01lS

^,

^{if f is c/}osed (resp. opell). Thcll f is c/osed (resp. open)

Proof lt i8 obviollS

Othcr rclationship between (Y

,

a) and (Y

,

δ) which is llseflll to charac- terize RS-continuity given in the following lemma.

Lemma 4.10. If(Y

,

ã) is HausdorjJ, then (Y

,

a) is RS-compact

P1"00f Lct {U" Q E \7} be a rcgular semi-opcn cover of (Y,a). Sinc~

(Y

,

^{δis IIallsdorff}

,

there cxist open disjoint sets U ancl V having RS- compact complemc따 Let {U"(i)

,=

1

,

2

,

3

, ... ,

n) and {Uß(i) : i = 1

,

2

,

3

,' . .

m} be finitc 8ubcollections of {U" α E \7} sllch that Y \ U c U，띠 (U"(i)) ancl Y \ \1 C Uint(U"(i)). Thus

,

{U"(i)

,

1

,

2

,

3

, ..

^‘

,

n}

U{U，β(i) ^‘ i

,

2

,

3

, ... ,

m} i5 a finite sllbcollection of {U,,: Q E \70} sllch thal.

Y = (Y \ U)U(Y \ V) c int((UU，α(‘) U UUß(i)) showing Y is RS-compact^‘ Theorem 4.11. F'o,. a funclioll f : X • (κ 0) ， if ( κδ) is rra^1lsdo,jJ

Then f is cOlltinuous ijJ it is RS-continuous

P7'00f 13y the abovc Icmma

,

wc conclllde that a = ã

,

ancl 80

,

the proof is trivial

5.

RS-continuity and Separation Axioms

The innuence of RS-continuity on some separation propcrtics are in- vestigatecl. Some results in [1 이 are i mprovecl

The following lemma will be used throughout this notc

Functions Based on Compactness 283

Lemma 5.1 [19]. Lel f : X • Y be a closed funclion

,

given V C Y and U E r(X) conlaining f-'(V). Then there exits W E r(Y) containing \1 such lhat f-'(W) C U

The following three results are improvement forms of Theorems (3.1)

,

(3.2) and (3.3) in [10]

Theorem 5.2. For a c/osed RS -continuous function f from normal space X onlo a space Y. The d;sjo;nt c/osed sμbsets ωhich are RS -compact of Y may be sepamled by open sets

Proof Lct Fi C Y

,

i = 1

,

2 be c10sed and RS-compact such tha.t F

,

^{nFì =}

ø

, then f-'( 낀)，; = 1,2 a.re disjoint c10scd sets in X. By normality or X there cxist U‘ E r(X), i

=

¹,2 such that U,

n

U2

= 0

, and f-'(낀) C Ui

i

=

¹,2. By Lcmma (5.1), there cxist 끼 E r(Y) containing Fi, i

=

¹,2’ , such that f-' ( κ )CUi^{， i=} I ， 2. Hcncef-'( 끼)， i = 1,2 are disjoint Theorem 5.3. lf f : X • Y is a c/osed RS -conlinuous suηective func- lion and X is normal

,

then Y is IJausdorff iJJ it is 7~

Proof One direclion of this Theorem is clcar. To show the sccond one

,

let y

,

^{~ Y2 bc any two} points of a T

,

^{-space Y, then by Lcmma (5.1)}

{ι }， i

=

¹

,

2 are c10scd RS-compa.ct. Thus f-'{Yi},i

=

¹

,

2 are c10sed scls in X. Since X is liorma.1 therc exist disjoint Ui E r(X) containing f-' {y;}, i = 1,2. ßy Lemma (5.1), there cxists 11; E r(Y) containing yi and f-' ( κ)

c

U" i E {1, 2}. Hence f-' {Y‘}

c

^{f-'( 끼)}

c

U‘’ , iE{1,2) Evidcntly, \1,

n

\12 =

ø

, and so, Y is Hallsdorrr.

Theorem 5.4. The inverse image of each T， -sψace uπder an injeclion RS-continuous function is T

,

Proof Lct f : X • Y bc injection RS-continllolls

,

Y be '1', and x E X

,

then {J(x)}

c

^{Y is c10scd and RS-com}pact [See Lcmma (3.4)]. Therefore {x} is c10scd in X. I-Ience X is T

,

Theorem 5.5. For any funclion f : X • Y

,

^ψe IllIve

(i) If f is RS-conlinuous and A C X

,

then the

,

'estreclion funclion flA ;s RS-continllous

(ii) [f {U" : ^Q E \7} is an open cover of X and f" = f j U" is RS continuous for each ^Q E \7, then f is RS -conlinuous

Proof (i) Let \1 E r(Y) having a.n RS-compact complement

,

by Theorcm (4.2)

,

we get f-'(\I) E r(X). Since UjAt'(\I)

=

^f-'(V)

n

A which is open in A

,

then f jA is RS-continuous

284 M. E. Abd EI-Monsef, R. A. Mahmoud and A. A. Nasef

(ii) Take V C Y is open and (Y \ V) is RS-compact

,

then J-'(V) =

u

{f;'(V) ‘ a E \7}. By RS-continuity ofeach Jo

,

^{weget J; '(V) E} T(U_o)

,

and so

,

J; '(V) E T(X). Therefore

,

J-'(V) is open in X

,

and hence J is RS-continuous

Lemma 5.6. For an open weakly-continuous surjective Ju πction J : X • Y the image oJ each open set having RS -compact complement relative to X is open having RS -compact complement πlative to Y

Proof Obvious by using Proposition (2.8).

Corollary 5.7. !J J : X • Y is an open O-continuous surjective Junction, then J(W) E 7"(Y) and Y \ J(W) is RS -compact l^'Elative to Y Jor each open set W having RS -cornpact complement l'Elatiνe to X.

Theorem 5.8. J X • Y and 9 Y • Z aπ two 끼nctions， such that their cornposition is RS -continuous and 9 is open weaklν-continuo 7.ls

sU1jection. Then J is RS -continuous.

Proof Immediately by Lemma (5.6).

Theorem 5.9. Let J : X • Y and 9 : Y • Z be Junclions

,

then the Jollowing statements ve7'iJy

(i) !J J is continuo따 and 9 is RS -con띠21m따l

continuous

(ii) [J J is s^U1jective open (l'esp. closed) and 9 0 J is RS -continuol찌

then 9 is RS -continuous

(iii) J is a quotient Junction. Then 9 is RS-continuous i

fJ

9 0 / is RS-continuous.

Proof (i) Obvious.

(ii) Follows diredly by using the statement (b) (resp. (c)) of Theorem (4.2)

(iii) Necessity

,

obtained directly by (i)

Suffìciency

,

let W ε 7"(Z) having RS-compact complement^‘ Then (g 0 /)-1 (W) = /-1 (g-1 (W)) E T(X). Since J is quotient, then 9-1 _{(W ) E} T(Y). Therefore 9 is RS-continuous

6.

ç- c ontinous func tions

Throught this article

,

we will use the abbreviations comp. set

,

coun comp. set

,

H-c.set

,

N-c.set

,

S'c.set

,

s-c.set

,

RS-comp.set

,

p-comp.sct

,

and Lind. set for a compact set

,

a countably compact set

,

an H-closed set

,

an N-closed set

,

an S-closed set

,

an s-closed set

,

an RS-compact set

,

Functions Based on Compactness 285

a para∞mpact set

,

and a Lindelöf set

,

respectively.

We call that W E P(X) is an ç-set if ç E

n = {

^comp.

,

coun. comp.

,

H-c.

,

N-

c.,

S-c.

,

s-c.

,

RS-comp.

,

p-comp.

,

Lind. set }.

Definition 6.1. A function f : X • Y is called ç-continuous if for each x E X and each open set V C Y containing f(x)

,

where Y \ V is an ç-set there exists an open set U containing x such that f( U) C V.

It is clear in Definition (6.1) that if

ç

is a comp. (resp. coun. comp.

,

H-c.

,

N-c.

,

S-c.

,

s-c.

,

RS-comp

‘,

p. comp.

,

Lind.) set. Then ç-continuity is identical with C (resp. C'

,

H

,

N

,

S

,

s

,

RS

,

para

,

L)-continuity

The following theorems are obtained for ç-continuity by usual tech- niques and the proofs are thus omitted

Theorem 6.2. The following statements are equivalent f0^1" a function

f

^‘ X • Y

(i) f is ç-continuous

(ii) V E T(Y ) having ç -set complement

,

ç E

n ,

then f-' (V) E T(X) (iii) f-l(F) is c/osed in X

,

for each F is c/osed ç-set in Y

,

ç E

n

Theorem 6.3. Let f : X • Y be ç -continuous

,

ç ε

{

^comp.

,

coun. comp.

set) and f(X) C Y is c/osed. Then f : X • f(X) is

ç

-continuous

Theorem 6 .4.

lf

f : X • Y is a ç-continuous injection

,

ç E { comp.

,

N-c.

,

S-c.

,

s-c.} and Y is a T

,

-space. Then X is T₁

Theorem 6.5. If f : X • Y 염 ç -continuous

,

^c/osed surjective and ç E { comp.

,

N-c.

,

S-c.

,

s-c.

,

RS-comp. } from a normal space X into a T

,-

space Y then Y is Hausdor

ff.

Theorem 6.6. If f : X • Y is an open function with a c/osed graph

,

then f is

ç

^-conti1l1

‘

^ous

,

ç E {H-c.

,

S-c.

,

s-c.}

Theorem 6.7. If f : X • Y is a c/osed

,

ç-continuous

,

ç E {S-c.

,

s-c.

,

p-comp. }, ψhere X is normal. Then d성:joint c/os떠， ιsubsets of Y may be separated by open sets.

Corollary 6.8. If f : X • Y is a closed

,

ç-continuos

,

ç E {S-c.

,

s-c.

,

p-comp. }. where X is normal

,

then Y is T₂ i

fJ

Y is T

,

^-space.

Theorem 6.9. If f : X • Y isç-continuous

,

ç E { comp.

,

H-c.

,

N-c.}, Y is Hausdor

fJ

and f(X) is an open subspace of Y. Then f : X • f(X) is ç-continuous

,

ç E { comp.

,

H-c.

,

N-c. }.

286 M. E. Abd EI-Monsef. R. A. Mahmoud and A. A. Nasef

Theorem 6.10. For a function f : X • Y

,

the following resu^/ts are hold if ( E { comp., coun comp., H -c., N-c., S-c., s-c., RS-comp., p-comp., Lind. set }

(i) If f is ç-continuous and A C X

,

then f /A is ç-continuous

(ii) Jf {U_o : <> E \7} is an open cover of X and ν fOI. each ^<>, f /U_{o is}

ç

-continuous

,

then f is

ç

-continuous.

Theorem 6.11. If f : X • Y

,

g: Y • Zaπ functio따 and ( E { comp.

,

coun comp., II-c., N-c., S-c., s-c., RS-comp., p-comp, Lind set.}. Then,

(i) 9 0 f is ç-continuous

,

if f is continuous and 9 is ç-continuous

(ii) 9 is

ç

-continuous, if f is open surjection and 9 0 f is

ç

-continuous (iii) Jf f is a quotient function

,

then 9 is

ç

-continuous i

.ff

9 ⁰ f is a/so

ç

-continuous

ACKNOWLEDGEMENT. Thc allthors wish to cxprcss th^í'ir deep gratitlldc to the referce [01" his valllable comments and sllggC'stions which have organized th is papcr

References

[IJ Gentry, K. R. and Iloyle, H. 13 ‘ C-continuous fU^l1ctions, Yokohama Math. J., 18(2) (1970), 71-76

[2J Park, Y. S., C' -contin

’‘

ous junctions, J. Korean Math. Soc. 8(197]), 69-í2 [3J Long, P. E. and lIamllctt, T. R., H-continllolls rnnctions, ßull. U.M.I., 2(4),

(1975^), 552-559

[4J Kohli, J. K., A class 01 mappings containing a^/l continuous mal'lJings, Glasnik Matematickl, 16(36) (1981), 361-367

[5J Long, P. E. and lIerrington, L. L., Para-conlinuous ηmctw ’‘ 5， Proc. Math. Phys Soc. Egypt, 52(1981), 1-5

[6] Malghan, S. R. and Hanchinal11ani, V. V., N-conlinuollsfullctiolls, Ann. Soc. Sci IJrllxelles, (98), (1984), 69-79

[7J Abd EI-Monscr, M. E. and Mahmolld, R. A., NC ,wd o'-cont^l1luous ^(I'"111';1I9S, Proc. of O.R. and Math. Methods Con. (1985), Bull. I'ac. Sci. Alex. Univ. Vol (26A) (1) (1986),28-38

[8} Di. ^l\'Ia.io. G., S-c/osed space, S-sets and S-continuous funclio^l1s, Att.i-Accacl ScÎ Torino (118), (1984), 125-134

[9J Lashin, E. R., O^{Jl Jl}o^Jl-conti^Jluous mappings, M.Sc. Thesis, Zagazig Univ. (1986)

Functions 8ased on Compactness 287

[10] Abd EI-Monsef, M. E., Mohmaud R. A., Nasef, A. A., s-cntinuous finctions, Proc 7th symp. Gerne때ra띠ITop。이logy & i

“

^{App미띠licκca따tìon。αon，’ M’삐arns。야ra} Univ. 25-27,’ July (1989잉 )

[1끼비1I1] 1Iam마1

t

“’

^te

“

^{s in Gen}

‘“

^{era/ Top%gy,} Amer. Math. Soc. 3(1981), 1-68

[12] Noiri, T., On RS-compact spaces, J. Korean Math. Soc. 22 (1985), No. 1, PP 19-34

[13] Levine, N., A decomposition of contim

“

^{ty in} topological spaces, Amer. Math Monthly, 68(1961), 44-46

[14] Fomin, S. V., Extensions of top%gica/ spaces, Ann. of Math. 44(1943), 471-480 [15] R∞e ， D. A., Weak-continuily and almost-continu씨， (preprint)

[16] Mrsevic, M. and Reilly, 1. L., On N-continuity and co N-c/osed top%gies, (t。

appear in ^Richeshe di Math.)

[1 7끼7끼] Noiπri1，’ T ’ N -c/05ed 5e

“

and almost c이/0야5e뼈dma‘'"때pp끼'"’”‘g5’ Gαla잃SI1l싸l 341-345

[18] Long, P. E., Hendrα ， M. 0., Properties 01 C -continuous functÎons, Yokohama Math. J., V. (22) No. (1-2) (1974),117-123

[19] Dugundji, J., Top%gy, Prentice I-Iall, India (1975)

DEPARTMENT OF MATIIEMATICS, FACULTY OF SCIENCE, ^{TANTA UNIVERSITY}, ^ECYPT

DEPARTMENT OF MATIIEMATICS, FACULTY OF SCIENCE, ^MENOUFIA UNIVERSITY,

ECYPT

DEPARTMENT OF MATHEMATICS, FACULTY OF SCIENCE, TANTA UNIVERSITY, ECYPT