615
Internat. J. Math. & Math. Sci.
VOL. |6 NO. 3 (1993) 615-616
A TYCHONOFF NON-NORMAL SPACE
V, TZANNES
Department of Mathenatics
University of Patras
Patras 26110, Greece
(Received July 13, 1992 and in revised form February 9, 1993)
ABSTKACT. A Tychonoff non-normal space is constructed which can be used for tim constru,ction of a regular space on whicll every weakly continuous (hence every 0-continuous or
/-continuous) map into a given space is constant.
KEY WORDS AND PHRASES. Tydmnoff, non-normal, weakly, O-,I1980 AMS SUBJECT CLASSIFICATION CODE. 54D15, 54C30.
continuous maps.
1. INTRODUCTION.
We construct for every Hausdorff space R a Tycho.off ,ran-normal space S such that if f
is a weakly continuous map of S i.to R theu there exist two closed subsets K’, L’, K’
such that f(K’) )r(L’) {r}, r R. Therefore, applyi.g the ,nethod of Jo.es [1], we
first construct a regular space containing two poiuts -oo, + oo such that f(-oo) f(+oo), f)r
every weakly continuous ,nap jr of this space i.to R an4 then, applying the method of llia, lis
and Tzannes [2], a regular space on whid every weakly co.tinuous (hence every 0-continu.ts
or r/-continuous (Dickman, Porter and Rubin [3])) ,nap into//is co.sta.t. The co,mtructiou
of S is a modification of the space T (R) in lliadis a,d Tzannes [2]. For regular spaces on which
every continuous map into a given space is constant see also Armentrout [4], Brandenburg and
Mysior [5], van Douwen [6], Herrlich [7], Hewitt [8], Tzannes [9] and .Jounglove [10]. A map
jr X -, Y, where X, Y are topological spaces is called 1) weakly continuous if for every
such
6 X and U open neighbourhood of ]() there exists an open neighbourlK)od V of
that/’(V) C_ CIU, 2) 0-conti,mous if for every z 6 X atd open neighbourhood U of/(.T),
there is an open neighbourhood V of : such that jr(CIV) c__ CIU 3) r/-conti,mous if for every
,
regular-open sets U, V of Y,
(i)
f-’ (U) C_ intClf-’ (U)
(ii) I.tCI/-’(U c V) C_ IntClf-’(V) c IntCll-’(V).
Every N-continuous is 0-continuous (Dickman, Porter and Rubin [3, Proposition 3.3. (c)])
and every 0-continuous is obviously weakly continuous.
We denote 1) by ]X the cardinality, of X, 2) by p(X) sup{p(X,x) x X} the
that is the minimal
pseudocharacter of X, where (X,x) is tl,e pseudocharacter of X at
of
cardinality of pseudobases of x. (The set U. consisting open neigl,bourhoods of x, is called a
pseudobasis if fUa {x}), 3) by +(X) the snallest cardinal number greater than (X).
2. THE SPACE S.
Let R be a Hausdorff space and K, L two uncountable sets such that IKI ILl
.,
V. TZANNES
616
For every ki E K (reap. I, E L) we consider
at,
uncountable set K, (reap. Li) and a t M
foli= to: Every int =ing to Ki, Li
la. For every ki E K (reap. Ii E L)
tim
a b=ia of open tmigh=la
O() {] UC, (gp. O(ll) (li} u Oi), wJre
C,, Mt of all but finite humor of ekmenta of K,, rltively. r ery point m E M
a ia of open h=lma =e tim
O(m) {m} U P U O, wlmre P, 0 =gain all but
finite humor of elemen of tim mrs (&,(m) C I}, ((m) E I}, r=pectively, where I an
index t, [I] R a= gi =e otm-to-oim inapa of M oo K=, i, rpectively.
One can almw that tim epa S M Tyclm=mff d n-normal.
t J a ly continuous map of S into R Sin Igl > lUl, it folws that for me
g such tim, IK’I IICI ./(K’) {rt). t (: n 1,,...} be a
rl E glare exgs
ubk sut of Kt. Sin for ery open igh=l U of r tl t/-’ (CIU) ntait= an
open tmiglo=ld of n 1, 2, it folws that IK, /-t (r,)l (R, r,). ntly,
if
k tl o--o map of M o, K, tln I;t(K, /-’(r,))]
(R, rt) and
,
,,
,
’,
(R, rl). ating 1 tl sw for tl t we Im that r
ra E R tlmre kt L’ ,1’1 I1,I(’) {r=} sad a mble subt {I,
1,2,...}
su, tlmt if V k an open ighbo=ld of r= tln I. /-’(r=) (R, r=) s= Inm
IU’(. /-’(,))1 (,). ,,r. ,r u’ U(’(. /-,(,)) u (. /-,(,)))
n:l
titan M M’
t m E M M’ and CI be s cd =mishbourld of/(,n) su= that
r=, r= CIW. lmre e m= open ishurld O(,n) of,n su= that/(O(m)) CIW, while
or ery n ], 2, l(m) E/-=(ri), g,(m) E/-=(r=) whi= imi,ly that/(,n) rj
IU&I(K /-(rt))[
.
1.
2.
3.
4.
5.
6.
7.
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