I nt. J. Mh. M. Sci.
Vol. 4 No. 2 (1981) 407-409
407
RESEARCH NOTES
A CHARACTERIZATION OF PSEUDOCOMPACTNESS
PRABDUH RAM MISRA
I.M.F.
U.F.G.
597
Caixa Postal
Goinia- Go, BRASIL
74000
and
VINODKUMAR
T.I.T.
New Delhi, INDIA
(Received October 6, 1980)
ABSTRACT.
It is proved here that a completely regular Hausdorff space X is
pseudocompact if and only if for any continuous function f from X to a pseudocompact space (or a compact space)
,
Y, f #
is z-ultrafilter whenever
is a
z-ultrafilter on X.
KEY WORDS AND PHRASES.
Pedo_omloc, X, z-f,
1980 IdATHEITICS SUBJECT CLASSIFICATION CODES.
i.
z-u fLun.
Pr/mary 54099.
INTRODUCTION.
For notations and basic results one is referred to [i].
We only consider
here completely regular Hausdorff spaces.
Let f be continuous from X to Y.
denotes the z-filter {B
.
Z(Y):
,
Let # be a z-ultrafilter on X, then f #
f-l(B)#}
We
on Y and is known to be prime.
further know that a prime z-filter is contained in a unique z-ultrafilter,
,
A(f) denote the z-ultrafilter containing f #.
from 8X to 8Y sending # to A(f)#.
Thus we have a function (f)
The function f is called z-ultra if f
A(f)# for every z-ultrafilter # on X.
Let
,
408
2.
P.R. HYSRA AND VINOD
MAIN RESULTS
PROPOSITION.
A continuous function f from X to Y is z-ultra if and only
Y,
B
Let f b,e z-ultra.
PROOF.
B--BY
f*# e
But this is equivalent to B e f *
if and only if
e f
Conversely, B e
f-l(B)
A(f) i
Then,
8X
-i
We see that f
or to f
if and only if
-I
(B)
,
A(f)
which happens
X
(B)
,
f
if and only if
A(f)-I[B--SY).
,
(B--BY)
But A (f)
e
--BY
B
f-l(B)
8X
--BY,
i.e. A(f) e B
since
is equivalent to saying that B e
(f).
A(f).
In order to prove the main theorem of the paper we need the following
observations for pseudocompact spaces.
If X is pseudocompact, then a subset of
8X is a zero-set if and only if it is closure of a zero-set in X and conversely,
a subset of X is a zero-set in X if and only if its closure is so in 8X.
If a space X is pseudocompact then any continuous function f
THEOREM.
from X to any pseudocompact space Y is z-ultra.
Conversely, if the inclusion
of X in 8X is z-ultra, then X is pseudocompact.
PROOF.
Let B be a zero-set in Y.
A(f)-I(B--BY)
A(f)-I(B--BY) --A--Sx for
is a zero-set in
B
Since --BY
pseudocompact,
is a zero-set in 8X.
that
some zero-set A in X.
Since
X
Pseudocompactness of X implies
X
A.. Hence
(f)-l(B)
f-l(B) A,
X
(f)-l(B) X f-l(B).
f-l(B). Next, A(f)-I(B--BY)
Clearly, A(f)
X
-I
--Sx
A
and we have f to be z-ultra.
Conversely, let i be the inclusion of X in 8X.
ls the identity on 8X.
f-l(B).
We show that A
A(f)/X-- f, we observe that A
--BY) C
(B
8Y as Y is
Since
Let B be a nonempty zero-set in 8X.
A(i)/X
i, A(i)
Slnce i is z-ultra,
X
X
from the above proposition we have that B
and
[1,61.I] shows that X
(1)-I(B) I-I(B)
B
X
is pseudocompact.
As an appllcatlon of our theorem we prove the following well known theorem
due to Glicksberg [2].
409
SHARACTERIZATION OF PSEUDOCOMPACTNESS
If X is pseudocompact and Y is compact, then X x Y is pseudo-
THEOREM.
compact.
PROOF.
space.
Let f: X x Y
/
Z be a continuous function, Z some pseudocompact
Consider a z-ultrafilter
0
on X x Y.
2
projection on the second coordinate.
it is fixed as well.
subspace X x
{yo }
Let
,
f
,
f
0
fil(B),
filter.
,
Since Y is compact and
20.
Yo
Hence
101 Next, let B
f-l(B) intersects
We get,that B
{yo }.
f
Hence
every member of
,
f 0.
Let
fl
Since X is pseudocompact,
101
Hence
,
f
.
fl(B)
f
Thus
ii_
+
,
Y denote the
2
is a z-filter,
the restriction of
01
{yo }.
is a z-ultrafilter on X x
of f to the subspace X x
X x Y
Let
0
to the
denote the restriction
fl
is z-ultra.
Clearly,
01. Since f-l(B) contains
f-l(B) 0 as it is a z-ultra-
and it follows that f is z-
ultra.
ACKNOWLEDGEMENT
This work was done while the first author was visiting Mehta Research
Institute, Allahabad in summer 1977.
REFERENCES
I.
Gillman, L. and Jerlson, M., Rings of Continuous Functions.
Princeton, 1960.
2.
Gllcksberg,l., Stone-Cech Compactifications of Products, Trans. Amer. Math.
Soc. 90 (1959), 369-382.
Van Nostrand,