On the infimum of a family of topologies for a set

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE X I (1967)

N. L

(Columbus, Ohio)

On the infimum of a family of topologies for a set

If X = X x yr% where X x and X % are topologies for a set X, then in general, Ж ( х , Х ) Ф Ж(х, Х г) гл Ж (ас, Х 2) where Ж denotes neighbor­

hood system. To see this, let X = {а, b, c} , X x = ( 0 , w , LI and У 2

= j0, {a, b}, X). Then

{а,Ь}еЖ{а,Хх) гл Ж ( а , Х 2), but {а, Ъ}4Ж(а, X ) .

In this paper, c denotes the closure operator, В the derived-set operator, Int the interior operator, the complement operator, r the rim operator (rA = A — Int A) and b the boundary operator. A always denotes a nonempty index set.

T

1. Let X a be a topology for X for each aeA and let X = (~}{Xa: aeA). Then the following are equivalent:

(1) Ж ( х , Х ) — (~){Ж(х, X a): aeA } for each x in X,

(2) if Oae X a for each aeA, then there exists an Oe X such that P|{Oa:

aeA}

О Я ( J { O a : aeA},

(3) for each A s X, B A = [J { B aA : aeZl}, (4) for each A я X, cA = {J{caA: aeA}, (5) for each А я Х , rA = \J{raA: aeA}, ( 6 ) for each А я X, bA = {J{baA: aeA}, (7) for each А я X, Int A = P { I n tadL: aeA},

( 8 ) if F a

C € X a for each aeA, then there exists an F in ФХ such that П ( Р . : a < J ) ę # ę U R : « 4

Proof. (1) -> (2). Let Oae X a for each aeA. If P {Oa: aeZl} = 0 , take О = 0 . If f ) {()a: aeA} Ф 0 , let же p { 0 a: aeA}. Then (J {Oa:

ae А}еЖ(х, X a) for each aeA and by (1), P ( 0 a: ae А} еЖ(х, X) . Thus there exists an Ox e X such that xe()x я p [Oa: aezJ}. Let O — P [Ox:

xe П {Oa: aeA}).

(2) -> (3). Let A я X. Since я 9~а foi each aeA, it follows that B A я DaA for each at A. It suffices to show then that DA я (J {DaA:

aeA}. Suppose х ^ В аА for each aeA. Then for each aeA, there exists an 0 aeć7~a such that xeOa and 0 a ^ A я {x}. B y (2), there exists an O e J such that П { 0 a: aeA} g 0 ę U {0 a: aeA). Hence xeO and 0 r\ A

^ U {ОаГ, A: aeA} я {x}. Thus x^DA.

(3) -> (4). Let А я X. B y (3), DA = P {BaA: aeA} and hence cA = A 'u B A = A

P {B aA : aeA} = (J { i u B aA : aeA} = p {caA : aeA}.

(4) (5). Let А я X. B y (4), cVA = p {caVA: aeA}. Then rA — A r\ ^ in tH — A r\ cVA = i n U {caVA: aeA}

= pj }A гл caVA: aeA} = yj{raA : a e A } . (5) -> (6). Let А я X. Then

ЬА = cA гл cVA = cA гл c&A ^ (A w VA) = A rs cVA

У?A r\ cA

= rA

rVA = (J {raA: aeA} w p {raVA: aeA}

= p {raA w raVA: aeA} — P {baA: aeA}.

(6) -> (4). Let А я X. Then

cA = A w ЪА — A ^ p {baA : aeA} = P {A w baA : aeA}

= U К -4.: aeZl}.

(4) -> (7 ). Let А я X. Then

In tH = VcVA = V U {CctfA: aeA} = P {^са<ГА: aezl}

= P {IntaH: aeZl}.

(7) -> (1). I t suffices to show that p {

Y {x,tXa):

g

for each x e X . Let N e p { ^ ( x * , ćXa) : aeA}. Then a?*elnta7\r for each aeA. B y (7), a?*elnt7\r and hence ХеЖ(х*, 3~).

(2) - p 8 ) . Let F ae V ^ a for each aeA. B y (2), there exists an O e J such that P | {VFa: aeA} я О я [ J {VFa: aeA}. Taking complements,

П { Fa: aeA} я VO я U {Fa: aeA}. Let F = VO.

(8) -> (2). We leave this as an exercise for the reader.

T

2. Let a be a topology for X for each aeA and let F ~ P { T a: aeA}. Suppose further that Y я X , Wa — Y r\ 3~a for each aeA,

°ll = P {/?/„: aeA} and J /'(x,^~) = P { ^ ( x , ^ a): aeA} for each xe X.

Then

(1) Щ — Y rs'T

and

(2) П { Л (у ,т .у . а<Л} = Ж ( у , « Г )

for each у e Y.

P roof. (1) Let Ue%. Then UeWa for all aeA and hence there exists an Oae X a for each a such that U — Y ^ 0 a. Then U £ P| {0 a: aeA}.

By (2) of Theorem 1, there exists an O c J such that П {Oa' aeA} g O g U {0a: aeA}.

Thus

Then

P ę Y n O g Y n U {Oa: aeA} = U ( ^ ^ 0 «: aeA} = U.

U = Y гл О and U e Y г л У .

Conversely, let TJ e Y гл X . Then U = Y r\ О where О e X . Since X ę £Ta for all a,

U e Y 1?~a = 91 a for all aeA.

Hence U e 91.

(2) We use (4) of Theorem 1. Let c* denote the closure operator in {Y, 99} and c* the closure operator in {Y, 9Ja}. Then if P ę Y, c * P = y n cP = Y n U { c . P : a e A } = U { Y r ^ c aP-. aeA} = \J{c*aP: aeA}.

In connection with Theorem 2, it seems appropriate to consider the following example: Let X , X'1, X 2, ^ ' be as in the example at the beginning of the paper and let Y = {a, c}. If 99x = Y r\ ,9l2 =■ Y r\ «^"2 and 99 — 9lx r\ 99 2, the reader can easily check that 99 Ф Y ^

T

3. Let f: X ->■ Y be a single valued transformation and let 99a be a topology for Y for each aeA; let 99 = P) {99a: aeA}. Further­

more, let X a = f ~ 199a for each aeA and ЗГ — p {Xa: aeA}. I f Ж(у,9/)

= П 99 a): aeA} for each y e Y , then

( 1 ) r = t lt!

and

( 2 ) J f { x , X ) = П { ^ ( х , Х - а): aeA}

for each x in X.

Proof. (1) Let OeX. Then O e Xa = f ~ 199a for each aeA. Thus О — f ~ 1Ua for some Uae99a for each aeA. By (2) of Theorem 1 , there exists a C e f such that P) {I7a: aeA} ę U £ U {?7a: aeA}. Now

^ = П { T ' V u . a e A } = f ~ 1n { U a: aeA} s f ~ l U £ U {f~l U a: ae A} = 0.

Thus О — f ~ l U and hence У s p 1*?/. Conversely, let 4 e / _ i t . Then

A e/~ 1 99 a for each aeA. Hence A e X a for each aeA which implies that

A e X . Thus f ~ x99 £

(2) We employ (2) of Theorem 1. Let 0 a€ ^ a for each aeA. Then 0 a = / ~ 1 TJa where и ае ^ а . There exists a Ve°U such that

П { и а: а е А} я U Я ( J { Ua: aeA}.

Now

П { 0 „: aeA} = Г К Г 1^ :

= Г 1П l^«: s f ^ s U (T 1^ : = U {Oa: aezl}.

Let О = t 1U.

In regard to Theorem 3, the following example is pertinent: Let Y = {a, b, c}, °llx = (0 , [a], Yj, % = {0, {a, b}, Y} and °tl = °UX rs Let X = {a, c} and f{a) = a ,/(c ) = c. Then if ■T1 = / _1^,1 and <Y"2 = / _1‘$f2) then where F r\ ЗГг . We leave the details to the reader.

L

1. Let f : X -> Y be a single valued transformation. For each aeA, let X a be a topology for X and X — f } { T a: aeA). I f = {U:

f ~ 1U e X a} for each aeA and °U = Q [%a\ aeA}, then

<W = { JJ: f - YTJe«T}.

P ro o f. V е<Ш iff TJe°Ua for each aeA iff f ~l TJeXa for all aeA iff f - ' U e X .

T

4. Let f: X Y be a one-to-one single-valued transforma­

tion and let X a be a topology for X for each aeA. Let Wa = {U: t ' U e X a } for aeA and

I f — f"') { X a: a eA }, — P) {Wa: aeA}.

I f

JA{x,&~) = P) {

Y (

, X a): aeA} for each x e X ,

then for each у in Y,

■yY{y,W) = П { ^ ( y , Wa)- aeA}.

P ro o f. Since я °lla for all aeA, it follows that

•N’i y, ®) S П { X { y , ^ a): aeA}.

Conversely, let I e / ( i / , f a) for each aeA.

C ase 1. If y i f X , then f ~ x{y} — 0 e Y . Then by Lemma 1, {y}z°tt and N еЖ( у, <%).

C ase 2. If y e f X, then for some x in X, f(x) = у e Ua £ Ж for some

UaeiWa and each aeA, and xef ~l Ua £ / -1 W. Thus for

each aeA and hence f ~ l N e

There exists an O e X such that

xeO £ / _1W and у = f { x ) ^ f ( O) я N. Since / is one to one, О = f ~ xfO

and hence fOeW. From this it follows that

T

6 . Let and be topologies for X in which each closed set is a Gd. I f У — X x rs ST 2 and

= П { ^ ( ^ ^ г ) : i = 1 , 2 }, then relative to fX, each closed set is a Gd.

Proof. Let F be a ^-closed set in X. Then F is ^-closed and thus F = П {^S1): i ^ 1} and F = П (0{2): i > 1} where O p e ^ x and O p e ^ 2.

We may assume that 0 $ x £ 0 11} and Oflx £ Op for i > 1 . Using ( 2 ) of Theorem 1, there exists an OieX snch that OP гл OP g Oi ę 0 |J) w Op for each г > 1. Clearly F £ П {Ox: i > 1}. To show that (~) (0*: i > 1 } £ U, suppose Then and x 4 0 {^ for some ix and i 2. It follows then that x 4 OP w 0 | ^ for ъ ^ ^ “j % < £ 9

H (1 hence cc ^ | 0 | ^ ^ O f: i > 1 }.

Finally then, x i f \ {Ox: i > 1 }.

T

6 . Letf ^ '1 and ZT 2 be second axiom topologies for X and let ^ — !TX rs У g. I f Jf(x, ЗГ) = P) {J f( x } ^ i) : i = 1, 2}, then is a second axiom topology for X.

Proof. Let (Op: i ^ 1} be an open base for 9~x and {Op: i > 1}

an open base for У 2. Using ( 2 ) of Theorem 1, for each i and j, there exists an such that Op rs Op £ 0 # £ OP ^ Op. Now suppose x e O e ^ . Then 0 i = 1, 2, and thus there exist OP and Oj2) such that xeOP £ 0 and xe op £ 0. Then xeOy £ 0 and {Oijii,j ^ 1 } is a countable open base for

T

7. Let -Tx and X 2 be first axiom topologies for X. I f X == $~x rs X 2 and Xr {x,X') = P) {‘X'ix, X'f): i — 1 , 2 }, for each x e X , then fT is a first axiom space for X.

Proof. Modify the proof of Theorem 6 .

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