In paper [2], F. В. Jones has given a sketch of a simple proof of a theorem being Jnhasz generalization for an arbitrary infinite cardinal number т of Arhangelskii theorem; if a Hausdorff space X

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A N N A L E S SO C IE TA T IS M A TH EM A T IC AE P O LO N AE Series I : CO M M ENTATIO NES M A TH EM A T IC AE X V I I I (1975) R O C ZN IK I P O L SK IE G O T O W A R Z Y S T W A M A TEM ATY CZN EG O

Séria I : P R A C E M A T E M A TY C ZN E X V I I I (1975)

W. I

ytjlpa

and U. L

okek

(Katowice)

A remark on Jones’ proof of Arhangelskii theorem

In paper [2], F. В. Jones has given a sketch of a simple proof of a theorem being Jnhasz generalization for an arbitrary infinite cardinal number т of Arhangelskii theorem; if a Hausdorff space X is r-compact and every point xe X has a base of cardility not greater than r, then cardinality of X is not greater that 2T.

, In this note we want to notice that from Jones’ proof a generalization follows of another Arhangelskii theorem; if a regular space X is т-compact, t(X) < r and every point x e X can be obtained as an inter

section of not more than r open sets, then cardinality of X is not greater than 2T. Because some indications in Jones’ paper (see Lemma 3 in [2]) are not quite clear, all proofs in this note will be given in full details.

1. Let A be a topological space and s and r be infinite cardinal num

bers. Introduce some notations and definitions.

A space X is r-compact iff every open covering of X has a subcovering of cardinality not greater than r. A symbol t{X) < r means that for every set A cz X and for every point же cl A there exists a set T c A with |T| < r such that X€ cl I 7; с1тж < r means that for every set A cz X with \A\ < r is

|clA| < r;

w

(

s

, X ) < r means that every closed set i с I with \A\ < s may be obtained as an intersection of no more than т open sets.

Let coz denote the least ordinal number among ordinal numbers of power greater than r. Since r is infinite, a family of all limit numbers less than o T is cofinal in {a : a < wT) .

L

emma

1. Let t(X) < r. I f a family M cz 2х has the property:

() {for every N cz M with |A| < r, there exists N ', N a N' a M, such* that u l ' is closed in X ).

Then и M is a closed subset of X.

Proof. If же cluJf, then there exists a set T cz

kj

M such that |T| < r and a?e cl T. Since \T\ < r, there exists N c Jf \N\ <

t

, such that T с и A.

From property () it follows that there exists a family N' a M such that*

T c u X ' and clu A ' с и M. But же cl T cz c l u A ' с: и Ж Hence же u l .

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214 W. Kulpa and U. Lorek

Le m m a

2. Let a space X be such that t(X) < r and cl2r X < 2r. Let { Uy:

y < /3}, (3 < <üT, be a decreasing sequence of closed subsets and let { V T y: y < /?}

be a family of closed sets such that Vх c= Uy, \Vy\ < 2r and for every y0< (3, U { V ry: y + y0} is closed. Put Up = П {Uy; У < P}-

Then, if Up Ф 0 , there exists a non-empty closed set V} cz Up with \Vp\

< 2T such that U { V х: у < ft} is closed in X.

Proof. Put 8 = . \ J { V x y: y < p ] and 8 ° = cl 8 \ 8 . We shall show that 8 ^° c Up. Let xe 8 °. Suppose that x 4 Up. Then x 4 ^UY for every у ^ y0 and for some y 0 < (3. Hence x 4 & \ J { V ry; у > y0} <= UyQ, and hence if xe cl$ and X 4 Up, then xe cUJ (F:* = U { V ry: У ^ 7o} ^ hence x$ 8 °. Thus 8 °

c :

Up.

Since |$| < 2T and cl2t X < 2T, we have |$°| < 2r. If Up ф 0 , then let V be a non-empty subset of Up such that 8 ^°

^с

^{V and |} F| < 2T. Put Vх

= cl V. Since claTX < 2T, we have |FJ| < 2T. A set W = U {FJ: у < £}

is closed because IF = cl 8 ucl V.

Le m m a

3. Let a space X be such that cl2rX < 2T and w(2r, X ) < 2 T.

Then there exists an inverse system {Pa, a < ooT} , where P a is closed covering of X , |PJ < 2 X, wUh maps f ap: P a-+Pp, a > (3, such that:

1° if UpePp, then Up = U fâp(Up) and, in addition, if \Up\ < 2 ^{X, then} t â ( U p ) = {Up}-,

2° if a is not of a form a = (3 + 1 , where (3 is a limit number, then every non-empty element of P a is of a form C\{Up: (3 < a], where {Up}p<ae limlP^:

P < «} ;

3° if a = /8+1 and /3 is a limit number, then for every UpePp there exists a set Urae /а / (Up) such that \Ura\ ^ 2r and Ux ac\Ua = 0 for every Ua efâp(Up)\{Ul} and a set Urau {U x y: f y>y^ ( U x y) = f a>y- i ( U x a) , y - 1 is a limit number} is closed.

Proof. Let us put P 0 = {X } and let usvassume that the coverings Pp and the maps fpy satisfying the conditions of the lemma are defined for every (3 < a. Let us define P a.

If a is not of form a = (3 +1, where /8 is a limit number, then we define the covering P a as in the point 2 of the lemma and we put f ay(Ua) = Uy for Un ^ C\{Uy: y < a], {Uv}y<aelim{Py: y < a } . Notice that \Py\ < 2T, because |a| < r and [Py\ < 2T for every y < a.

If а = /8 +1 and /8 is limit number, then we define P a as follows : Let

UpePp. Then Up = C\{Uy: y < {3}, where { Uy} y<pe lim{Py: y < /3}. From

Lemma 2 it follows that for a set IF = (J { Ux y: f y,y- i ( U x Y) = fp,y-\( Up),

у < (3, у —1 is a limit number} there exists a non-empty closed set Ux a c Up

such that I P„| < 2T and the set Z7„uIF is closed. If | Up\ < 2T, then we put

Ux a = Up. The assumption w( 2 x, X) < 2T implies that Ux a is an intersection

no more than 2r open sets Gt, te T. We define a closed covering Cup of Up;

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Arhangelslcii theorem 215

Си = { U l } u { ( X \ G t) n U pz te T}. Put P a = U {Ощ- Vfi'Pfi} and define the map f ay for f aP(Ua) = f Py{Up) if TJae C Up.

Th e o r e m.

If r-compact space X is such that t(X)

<

r ,

cl2TX < 2 T

and w{2T, X ) < 2T, then \X\ < 2T.

Proof. Let {Pa: a < cor} be an inverse system satisfying all conditions of Lemma 3. We shall show that for every point x e X there exists an a (a?) < (ot and U^x)e Paix) such that xe Ua{x) and | Ua{x)\ < 2T. There exists a { Ha^ax<<oxe Km{Pai a < cor} such that xe f^\{Uax: ax < cor}. Suppose that for every ax, \Ua \ > 2T. Let us consider a family { JJx a : ax < cor} such that fa,a-i{Ulx) = fa,a-i(Uax), where a —1 is a limit number. From Lemma 1 it follows that a set В = \J{Ura : ax < a>T} is closed. A family P = {X \ Ua : ax < coT} is an open covering of the set B. Since TJra c

and the family { Uax : ax < coT} is decreasing, there are no subcovering of P of the cardinality not greater than r. This contradicts that В is r-compact.

For every x e X let us choose a Z7a(x)

^{e P a (x)}

for some a(x) < coT, such that \UaM\ ^ 2T. The cardinality of the family \üaM: xe X } is not greater **than |U{P.: a < « , } | < 2 ' . 3 * = 2 * . Hence |X| < 2' - 2* - 2*.**

2. In this part of the note wé shall prove some corollaries of the theorem.

A symbol гсс1(1 , X ) < r means that every point xe X may be obtained as an intersection of no more than r closed sets being neighbourhoods of the point x. Notice that for Tx space X is w( 1, X ) < гос1(1, X ) and if X is regular, then w ( l , X) = wcl( 1, X ). A symbol w(xe X) < r means that for every point xe X the weight at the point x is not greater than r.

Lemma 4.

I f w(xe X)

< r,

then t(X)

< r.

Proof. Let xe d A and B(x) with \B(x)\ < r be a base at the point x.

For every U e B ( x ) , let x v denote an arbitrary element of TJглА. Put T = {xLr: TJeB{x)}. Hence x e c lT .

Lemma 5.

If a space X is a such that t(X)

^

r and

wcl( l ,

X)

^ r,

then cl2TX < 2\

Proof . For every point xe X , let D(x) be a family of neighbourhoods of x such that \D(æ) \ < r and {гг} = P){clt7: Tie B(x)}. Let A a X and and \A\ ^ 2r. For every x e cl A there exists a set I d , \T\ ^ r such that æe clT. Hence {x} — n{d{Ur\T): U e B ( x ) } .

Thus for every point же cl A there exists a family of subsets of A such that \SF\ < x and if

P e

then |P| < r and {ж} = n {clP : F e V}.

There are no more than (2r)T = 2r subsets of A of cardinality not greater than

t

. Hence there are no more than (2r)T = 2r such families Thus |clA| < 2T.

Lemma 6.

I f a space X is r-compact and w(

1,

X )

<

r, then w{

2T,

X )

< 2T.

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216 W. Kulpa and U. Lorek

Proof. For every x e X let D(x) witli \D(x)\ < f be a family of open neighbourhoods of x such that {x} = p){î7: UeD(x)}. Let А с X be a closed set. Notice that A = P ){X \ {y }: у 4 A). Let у 4 A. Since A is.

r-compact, there exists a family Q <= \J{D(x): x A } with IQ I < r and* A c uQ <=. X \ { y } . Notice that HJ{Z)(a?): A}\ ^ 2r. Hence there are no more than (2T)r = 2T such families Q.

Lemmas 4, 5 and 6 and the theorem imply

Co r o l l a r y

1. I f a space X is r-compact, t(X) < r, wcl( l , X ) < x r then \X\ < 2T.

Co r o l l a r y

2 (Arhangelskii in case r = K0). I f a regular space X is x-compact, t{X) < r mid гс(1, X ) < x, then \X\ < 2T.

Co r o l l a r y

3 (Juhasz). I f a Hausdorff space X is x-compact and w(xe X) < x , then \X\ < 2T.

References

[1] A. V. A r h a n g e lsk ii, On cardinal invariants, General Topology and its Rela

tions to Modern Analysis and Algebra III, Third .Prague Topological Sympo

sium 1971, Prague 1972, p. 37-46.

[2] F. B. Jones, The utility of empty limits, ibidem, p. 223-228.

SIL ESIA N U N IV E R S IT Y , K A T O W IC E

In paper [2], F. В. Jones has given a sketch of a simple proof of a theorem being Jnhasz generalization for an arbitrary infinite cardinal number т of Arhangelskii theorem; if a Hausdorff space X

W. I

and U. L

(Katowice)

A remark on Jones’ proof of Arhangelskii theorem

, In this note we want to notice that from Jones’ proof a generalization follows of another Arhangelskii theorem; if a regular space X is т-compact, t(X) < r and every point x e X can be obtained as an inter­

section of not more than r open sets, then cardinality of X is not greater than 2T. Because some indications in Jones’ paper (see Lemma 3 in [2]) are not quite clear, all proofs in this note will be given in full details.

1. Let A be a topological space and s and r be infinite cardinal num­

bers. Introduce some notations and definitions.

|clA| < r;

(

, X ) < r means that every closed set i с I with \A\ < s may be obtained as an intersection of no more than т open sets.

Let coz denote the least ordinal number among ordinal numbers of power greater than r. Since r is infinite, a family of all limit numbers less than o T is cofinal in {a : a < wT) .

L

1. Let t(X) < r. I f a family M cz 2х has the property:

(*) {for every N cz M with |A| < r, there exists N ', N a N' a M, such that u l ' is closed in X ).

Then и M is a closed subset of X.

Proof. If же cluJf, then there exists a set T cz

M such that |T| < r and a?e cl T. Since \T\ < r, there exists N c Jf \N\ <

, such that T с и A.

From property (*) it follows that there exists a family N' a M such that

T c u X ' and clu A ' с и M. But же cl T cz c l u A ' с: и Ж Hence же u l .

2. Let a space X be such that t(X) < r and cl2r X < 2r. Let { Uy:

y < /3}, (3 < <üT, be a decreasing sequence of closed subsets and let { V T y: y < /?}

be a family of closed sets such that Vх c= Uy, \Vy\ < 2r and for every y0< (3, U { V ry: y + y0} is closed. Put Up = П {Uy; У < P}-

Then, if Up Ф 0 , there exists a non-empty closed set V} cz Up with \Vp\

< 2T such that U { V х: у < ft} is closed in X.

Up.

Since |$| < 2T and cl2t X < 2T, we have |$°| < 2r. If Up ф 0 , then let V be a non-empty subset of Up such that 8 °

V and | F| < 2T. Put Vх

= cl V. Since claTX < 2T, we have |FJ| < 2T. A set W = U {FJ: у < £}

is closed because IF = cl 8 ucl V.

3. Let a space X be such that cl2rX < 2T and w(2r, X ) < 2 T.

Then there exists an inverse system {Pa, a < ooT} , where P a is closed covering of X , |PJ < 2 X, wUh maps f ap: P a-+Pp, a > (3, such that:

1° if UpePp, then Up = U fâp(Up) and, in addition, if \Up\ < 2 X, then t â ( U p ) = {Up}-,

2° if a is not of a form a = (3 + 1 , where (3 is a limit number, then every non-empty element of P a is of a form C\{Up: (3 < a], where {Up}p<ae limlP^:

P < «} ;

3° if a = /8+1 and /3 is a limit number, then for every UpePp there exists a set Urae /а / (Up) such that \Ura\ ^ 2r and Ux ac\Ua = 0 for every Ua efâp(Up)\{Ul} and a set Urau {U x y: f y>y^ ( U x y) = f a>y- i ( U x a) , y - 1 is a limit number} is closed.

Proof. Let us put P 0 = {X } and let usvassume that the coverings Pp and the maps fpy satisfying the conditions of the lemma are defined for every (3 < a. Let us define P a.

If a is not of form a = (3 +1, where /8 is a limit number, then we define the covering P a as in the point 2 of the lemma and we put f ay(Ua) = Uy for Un ^ C\{Uy: y < a], {Uv}y<aelim{Py: y < a } . Notice that \Py\ < 2T, because |a| < r and [Py\ < 2T for every y < a.

If а = /8 +1 and /8 is limit number, then we define P a as follows : Let

UpePp. Then Up = C\{Uy: y < {3}, where { Uy} y<pe lim{Py: y < /3}. From

Lemma 2 it follows that for a set IF = (J { Ux y: f y,y- i ( U x Y) = fp,y-\( Up),

у < (3, у —1 is a limit number} there exists a non-empty closed set Ux a c Up

such that I P„| < 2T and the set Z7„uIF is closed. If | Up\ < 2T, then we put

Ux a = Up. The assumption w( 2 x, X) < 2T implies that Ux a is an intersection

no more than 2r open sets Gt, te T. We define a closed covering Cup of Up;

Си = { U l } u { ( X \ G t) n U pz te T}. Put P a = U {Ощ- Vfi'Pfi} and define the map f ay for f aP(Ua) = f Py{Up) if TJae C Up.

If r-compact space X is such that t(X)

r ,

and w{2T, X ) < 2T, then \X\ < 2T.

and the family { Uax : ax < coT} is decreasing, there are no subcovering of P of the cardinality not greater than r. This contradicts that В is r-compact.

For every x e X let us choose a Z7a(x)

for some a(x) < coT, such that \UaM\ ^ 2T. The cardinality of the family \üaM: xe X } is not greater than |U{P.: a < « , } | < 2 ' . 3 * = 2 * . Hence |X| < 2' - 2* - 2*.

2. In this part of the note wé shall prove some corollaries of the theorem.

I f w(xe X)

then t(X)

Proof. Let xe d A and B(x) with \B(x)\ < r be a base at the point x.

For every U e B ( x ) , let x v denote an arbitrary element of TJглА. Put T = {xLr: TJeB{x)}. Hence x e c lT .

If a space X is a such that t(X)

r and

X)

then cl2TX < 2\

Proof . For every point xe X , let D(x) be a family of neighbourhoods of x such that \D(æ) \ < r and {гг} = P){clt7: Tie B(x)}. Let A a X and and \A\ ^ 2r. For every x e cl A there exists a set I d , \T\ ^ r such that æe clT. Hence {x} — n{d{Ur\T): U e B ( x ) } .

Thus for every point же cl A there exists a family of subsets of A such that \SF\ < x and if

then |P| < r and {ж} = n {clP : F e V}.

There are no more than (2r)T = 2r subsets of A of cardinality not greater than

. Hence there are no more than (2r)T = 2r such families Thus |clA| < 2T.

I f a space X is r-compact and w(

X )

r, then w{

X )

< 2T.

Proof. For every x e X let D(x) witli \D(x)\ < f be a family of open neighbourhoods of x such that {x} = p){î7: UeD(x)}. Let А с X be a closed set. Notice that A = P ){X \ {y }: у 4 A). Let у 4 A. Since A is.

r-compact, there exists a family Q <= \J{D(x): x * A } with IQ I < r and A c uQ <=. X \ { y } . Notice that HJ{Z)(a?): A}\ ^ 2r. Hence there are no more than (2T)r = 2T such families Q.

Lemmas 4, 5 and 6 and the theorem imply

1. I f a space X is r-compact, t(X) < r, wcl( l , X ) < x r then \X\ < 2T.

2 (Arhangelskii in case r = K0). I f a regular space X is x-compact, t{X) < r mid гс(1, X ) < x, then \X\ < 2T.

3 (Juhasz). I f a Hausdorff space X is x-compact and w(xe X) < x , then \X\ < 2T.

, In this note we want to notice that from Jones’ proof a generalization follows of another Arhangelskii theorem; if a regular space X is т-compact, t(X) < r and every point x e X can be obtained as an inter

1. Let A be a topological space and s and r be infinite cardinal num

() {for every N cz M with |A| < r, there exists N ', N a N' a M, such* that u l ' is closed in X ).

From property () it follows that there exists a family N' a M such that*

Since |$| < 2T and cl2t X < 2T, we have |$°| < 2r. If Up ф 0 , then let V be a non-empty subset of Up such that 8 ^°

^{V and |} F| < 2T. Put Vх

1° if UpePp, then Up = U fâp(Up) and, in addition, if \Up\ < 2 ^{X, then} t â ( U p ) = {Up}-,

for some a(x) < coT, such that \UaM\ ^ 2T. The cardinality of the family \üaM: xe X } is not greater **than |U{P.: a < « , } | < 2 ' . 3 * = 2 * . Hence |X| < 2' - 2* - 2*.**

r-compact, there exists a family Q <= \J{D(x): x A } with IQ I < r and* A c uQ <=. X \ { y } . Notice that HJ{Z)(a?): A}\ ^ 2r. Hence there are no more than (2T)r = 2T such families Q.