Internat. J. Math. & Math. Sol.
VOL. 15 NO.
(1992) 207
207
AN OBSERVATION FOR SIMPLE EXPANSIONS
DAVlO A. ROSE and DRAGAN A.
JANKOVI(
Dcl)art==wnt of IMathcnati(’.s, East Central Univcrsity, Ada, ()K 7,1820
(Received March 26, 1991)
1980 MATIIEMATICS SUB.I ECT CLASSIFICATION- 54A 10, 5dB05
If (X,r) is a topological space and AC_X, the simple expansion of
" by A is the topology ’(A)
{UU(Vf3A)IU,V’}. Let us say that a topological property is weakly finitely additive if whenever
closed subspaces A and B have the l)roi)crty then so does A UB. Recall that a property is weakly
(open) hereditary if closed (open) subspaccs inherit it. It is well known that certain weakly hereditary,
weakly finitely additive properties such as normality, paracompactness, and the LindelSf property, lift
from (X,’) to
(X,v(A)) if and only if (X-A,’I(X-A)) has the property, when (X,’), and (X,v(A))
are
regular [6]. As in [2], a property is contagious if whenever a dense subspace has the property then so
does the entire space. Without any need for regularity or any separation we have the following simple
observation.
Theorem. If P is open hereditary, contagious, and weakly finitely additive, then (X,r(A)) has P
if and only if both
(X,r) and (A,r[A) have P (where rlA is the subspace topology).
and ’(A)IB
riB for B
CI-A
A or X-A. Among others, properties satisfying the hypotheses of the theorem include Baireness [4],
resolvability, and separability. One consequence is that (R, ’(I)) is a Baire space and hence second
The proof is a consequence of the easy facts that AE’(A),
category if (R,’) is the usual space of real numbers and
classification hole in
CI.(A)A,
is the subset of irrationals.
This fills a
[7].
REFERENCES
1.
2.
3.
4.
C.J.R. Borges, On extensions of topologies, Candian J. Math., 19 (1967) 474-487.
J. Chew, Contagious properties, Mathematics Magazine, 47 (1974) 84-87.
J. Dugundji, Topology, Allyn and Bacon, Inc., Boston, MA, 1966.
R. Haworth and IL McCoy, Baire spaces, Dissertationes Mathematicae 141, Polish Scientific
Publishers, Warsaw, 1977.
6.
N. Levine, Simple extensions of topologies, Amerzcan Mathematical Monthly, 71 (1964) 22-25.
D.F. Reynolds, Simple extensions of topologies, Topology: Proceedings of the Memphis State
7.
University Conference, Marcel Dekker, Inc., New York, NY, 1976, pages 239-242.
L.A. Steen and J. A. Seebach, Jr., Counterexamples itt Topology, ilold, Rinehart and Winston,
5.
Inc., New York, NY, 1970.