A C T A U N I V E R S I T A T I S L O D Z I E N S I S FOLIA M ATHEMATICA 7, 1995 Helena Pawlak O N T H E T O P O L O G I E S G E N E R A T I N G Q U A S I - I S O M O R P H I C R I N G S O F C O N T I N U O U S F U N C T I O N S
In th is p a p e r we show th a t tw o T ychonofr topologies on X are id en tic al if an d only if the rings of real con tinu ou is fu nctio ns co rre-sp o nd in g to th em are quasi-isom orphic.
In m any papers th e authors investigated the relations betw een the topological properties of spaces and the algebraic pro pe rtie s of rings of real contiuous functions defined on these spaces ([2], [6]). M any m athem aticia ns studied the situations when, in a given space X for different topologies, the classes of real continuous functions were identical ([3], [4], [5]). In this context, it is purposeful to seek for as weak assum ptions (of algebraic n atu re ) as possible, concerning th e rings of continuous functions, whose fulfilm ent would q u ara n tee th e equality of th e topologies corresponding to them .
It is well know th a t, for different Tychonoff topologies, the classes of continuous functions, corresponding to them , are different; con-sequently, th e equality of the classes of continuous functions for th e Tychonoff topologies im plies the identity of these topologies.
Therefore, it is n a tu ra l in our investigations th a t we assum e th e topologies unde r considerations to be T31.
In this pa per we use the sta n d a rd notations (from m onograph [1]). I f / is a real function defined on X , then we denote Z ( f ) = {x G X : f ( x ) = 0}. Let (X , T |), ( X , T2) denote topological spaces. We shall use th e term inology, for exam ple, T \-neighb ourhood or T \- continuity, to m ake a distinction betw een the two topologies und er consideration.
We say th a t the rings C (T \), C ( T 2) of real continuous functions on ( X , T i), ( X , T 2), respectively, are quasi- isom orphic if there exists a m ap pin g 0 : C{ T\ ) —> C ( T 2) such th a t 0 m aps C ( T \ ) onto C ( T 2) in a one-to-one way and Z { f ) = Z ( 0 ( / ) ) for each f G C {T \).
Let T \ , T2 be tw o Tychonoff topologies on a set X .
T h e o r e m 1. T h e topologies T\ and T2 are identical i f and only i f the rings C (T \ ) and C ( T 2) are quasi-isomorphic.
Proof. Necessity. O f course, if T\ = T2, then C ( T i ) — C ( T 2), a nd so, C ( T \ ) an d C (T 2) are quasi- isomorphic.
Sufficiency. Let A2 denote the family of all sets A consisting of all functions / G C{ T2) such th a t f ( xy \ ) = 0 for some fixed poin t x&. T his m eans th a t th ere exists a one-to-one correspondence betw een sets of th e fam ily A2 and points; precisely:
A 6 A2 if and only if / ( x ^ ) = 0 for / G A.
R em ark th at: x& ^ %B if and °nly if A ^ B\ m oreover, if A, B G A2 and A ^ B, then A is not included in B and B is not included in A.
Define th e neighbourhood system for «42 in the following way: for C G *42i let B ( C ) consists of all sets of the form {c} U { A G A2 : t ( x a ) ^ 0} for some function t G C ( T 2) such th a t t ( x c ) ^ 0. It is easy to see th a t the collection {B ( C ) } c e A2 satisfies conditions (B P 1)-(B P 3) from m onograph [1] (p .28).
Let now A2 denote the topological space w ith th e topology gener-a ted by th e neighbourhood system { B( C) } c e A*-
Let us define tw o m appings as follows: t : ( X , T x) - + A 2,
by th e form ula $ (A ) = xa
-It is no t difficult to see th a t £ m aps X onto A2 and $ m aps A2 o nto X and both are one-to-one.
Now we shall show the continuity of <£. Let A £ A2, $(.A) = x a , and let W be an a rb itra ry T2-neighbourhood of xa- Since ( X , T 2) is a Tychonoff space, then there exists t G C ( T 2) such th a t t ( x A ) 7^ 0 an d t ( x) — 0 for each x $ W .
C onsider th e neighbourhood U of A of the form U = { A } U { P < e A2 : t ( x P ) ^ 0}
O bserve th a t $ ( U ) C W . Indeed, let Q G U. In the case when Q = A, the above inclusion is obvious. Let then Q G U an d Q ^ A. T hu s Q = { / G C ( T 2) : / ( x q ) = 0}; m oreover, t ( x Q ) ^ 0. This m eans th a t x q G W , and so, $ (Q ) = Xq G W . This ends th e proof
of the continuity of <£.
Now, we shall show th a t $ -1 is continuous. Let x G X an d let U be an a rb itra ry neighbourhood of $ - 1(x). Then
U = { / G C (T 2) : f ( x ) = 0 } u { A e A 2 : t ( x A ) i 0}
for some ¿ G C ( T 2) such th a t t (x) ^ 0.
From th e T2-continuity of t we infer th a t there exists a T2- neigh-bo urho od W of x such th a t t ( w ) ^ 0 for each w G W . T hen Q - ' i W ) C U.
Now, we shall prove th a t £ is continuous. Let x G X an d let U be an a rb itra ry neighbourhood of £(x). T hen
U D { / G C ( T 2) : f ( x ) = 0} U { A G A2 : ¿ (xa ) ^ °}
for som e t G C ( T 2) such th a t t ( x) ^ 0. Let 0 be a quasi-isom orphism from C ( T \ ) to C ( T 2) and let t = 0 ( i i ) where t\ G C( T\ ) . By the p ro p erty of 0 , ¿i(x) # 0. Let be a T \-neighbourhood of x such th a t ¿i(u>) 7^ 0 for each w G W . Observe th a t £(W ) C U . Indeed, let w G W and C(w ) = i f *= C ( T 2) : f ( w ) = 0} = P. Since
therefore 0 ^ Q( t i ) ( w) = t (w). T h us P 6 { A € A2 : ¿ (z a ) # 0} and, consequently, ( ( w ) G i/.
Now, we shall verify th a t the m apping £-1 is continuous. Let A £ A i a nd £- 1(.A) = xa- Let W be an a rb itra ry T\ - neighbourhood of x a- Since Ti is a Tychonoff topology, there exists a function £ C ( T \ ) such th a t
(x a ) ^ 0 and t i ( x ) = 0 for each x (f: W.
P u t t = 0 (< i). T hus
t ( x A) 7^ 0 and t ( x) = 0 for each x ^ W.
C onsider a neighbourhood U of A defined as follows:
V = { A} U { P 6 A 2 : t ( x p ) i 0 }.
T h en C W .
To com plete the proof, we rem ark th a t the com position h = $ o £ is a hom eom orphism .
Re f e r e n c e s
[1] R. E ng elk ing, G eneral Topology, Po lish Scien. P ub l., W arszaw a, 1977. [2] L. G illm an and M. Je riso n , R ings o f continuou is fu n c tio n s, S p rin g e r- V erlag,
New York H eidelberg B erlin, 1976.
[3] E. K ocela, P roperties o f som e generalizations, o f the n o tio n o f co n tin u ity o f a fu n c tio n , F u nd . M a th . 78 (1973), 133-139.
[4] B. K oszela, T . Św iątkow ski and W . W ilczyński, Classes o f co ntin u ou s real fu n c tio n s, R eal A nal. Exch. 4 (1978-79), 139-157.
[5] H. N onas, Stron g er topologies preserving the class o f co ntin u ou s fu n c tio n s, F u nd . M ath . C l (1978), 121-127.
[6] M .H . S tone, A p p licatio ns o f the theory o f B oolean rings to general topology, T ra n s. A m er. M ath . Soc. 41 (1937), 375-481.
Helena Pawlak
O T O P O L O G I A C H G E N E R U J Ą C Y C H Q U A S I - I Z O M O R F I C Z N E P I E R Ś C I E N I E
F U N K C J I C IĄ G Ł Y C H
W pracy tej zostało pokazane, że dwie topologie Tichonow a okreś-lone n a zbiorze X są identyczne wtedy i tylko wtedy, gdy odpow ia-dające im pierścienie funkcji ciągłych są qua.si-izomorficzne.
In s titu te of M a th em atics Lódź U niversity ul. B an ach a 22, 90 - 238 Lódź, P o lan d em ail: rpaw lak ® p lu n lo 5 1.b itn e t
