203
Internat. J. Math. & Math. Sci.
(1991) 203-204
VOL. 14 NO.
RESEARCH NOTES
PROPERTIES OF C-EXPANSIONS OF TOPOLOGIES
DAVID A. ROSE
Department of Mathematics
East Central University
Ada, OK 74820
1990)
(Received August 25, 1989 and in revised form February 2,
ABSTRACT. The results of O. Njstad for a-topologies together with the results of P. L. Sharma for
compact perfect Hausdorff spaces are combined to produce several counterexamples in one space.
a, ti-
KEY WORDS AND PHRASES. a-topology, nowhere dense set, crowded spae, anti-compact space.
1980 AMS SUBJECT CLASSIFICATION CODE. 54G20.
enlargement of the usual topology on the unit interval of real numbers, [0,1], several striking
a
counterexamples are obtained at once. O. Njstad [1] introduced and studied the a-topology r for an arbitrary
topological space (X,r). This topology can be defined as the smallest expansion of r for which the nowhere
By
an
dense subsets of X relative to r are closed. Njstad
and Andrijevi6
[2] found IntaClaA
[1] noted
Int Cl A for every A
that
a
C X where
{U
NIUer and
Int(Int a)
and
N is nowhere dense},
CI(CI a)
are the interior
a,
a
r(ra). Letting X be the space X with topology r it follows immediately that X
and X a have the same regular open sets, nowhere dense sets, meager sets, clopen sets, and semiopen sets. [3]
and closure operators for
Recall that a set A is r-semiopen if A C_ Cl Int A. By X s, the semiregularization of X, is meant the space X
a
vith topology r having the v-regular open sets as a base. Clearly, X and X have the same semiregularization.
Topological properties mutually shared by X and X a are called a-topological and arc known
the sclnitopological properties of Crossley and Hildebrand
[5]. Among
selmrabilit.y arc also .-topological sicc X a,d X a share tim
saint
these is Bairc,css.
[4]
to coincide witl
Rcsolvability
dcnsc sets (Propositio, 1). Of co,rsc, tlw
topological properties include the semiregular properties shared by a space and its semiregularization. Some
important semiregular properties are: Hausdorff separation, Urysohn separation, extremally disconnectedness, llclosedness, pseudocompactness, connectedness, and almost regularity. A space X is almost regular if and only if
X is regular. Clearly X is regular if and only if X is both semiregular (X
PROPOSITION 1. The spa=es X and X a share the same dense sets.
Xs) and almost regular.
PROOF: Since rC_r a, each dense subset of X a is dense in X. Suppose that D is dense subset of X and
Wra-{(}. Then W U-N for some Ur and N, nowhere dense. Since U-CI N (r-{{}, (U-Ci N)f3 D 1, so
that W
D =/= (. Evidently D is a dense subset of X F!
a.
By
a
theorem of P. L. Sharma
[6],
if the nowhere dense subsets of a crowded (without isolated points)
liausdorff space are closed then countably compact subsets of the space are finite, i.e. the space is anti-countably
compact in the sense of P. Bankston
Further, such
a space is
[7] and
therefore also anti-compact since compact subsets are finite.
nowhere locally compact
[8]
since interiors of compact subsets are empty.
To
use
Sharma’s result we note the following, whose proof is an easy exercise.
PROPOSITION 2. The space X a is crowded if and only if X is crowded.
By considering two-point Sierpinski space, we see that crowdedness (like Baireness, resolvability, and
separability) is an a-topological property which is not semiregular. In the remainder of this article [0,1] is the
D.A. ROSE
204
unit interval of real numbers with the usual subspace topology. We now have the following examples.
EXAMPLE 1. The space [0,1] a is a crowded, Ilausdorff, no,semiregular, almost regular, nowixctc locally
compact, Baire space.
in
EXAMPLE 2. The space [0,1]a is a separable Hausdorff space which is not a k-space.
EXAMPLE 3. The space [0,1] c’ is pseudo-compact but anti-countably compact.
a
a
Since the identity function from X to X is continuous, cmmcctcd subsets of X are connected in X. But
general, it is not known whether connected subsets of X are conncctcd in X a. However, the connected subsets
are the intervals which if not singleton arc semi-open. Therefore, by showing that for semi-
of the space
[0,1]
open subsets
A of X, (r[A) a
subsets of [0,1]a
ralA,
are precisely those of
from connectedncss being a-topological, it will follow that the con,cereal
[0,11.
PROPOSITION 3. For any subset A C_ X,
PROOF: If W(rlA) a, then W (U
Since N is nowhere dense in X we have that W (U N)
I. L. Reilly and M. K. Vamanamurthy [9] have shown that the reverse of the inclusion of Proposition
Iolds if A is semi-open in X. By the foregoing remarks, the connected subsets of [0,1] a are tim i,tervals. But
since
[0,1] a
is anti-compact, the continuous functions frotn
[0,1]
into
[0,1] a
arc constant since eacl las a
nonempty connected and compact inaage.
EXAMPLE 4. Intervals are connected in the totally path disconnected space [0,1] a.
ACKNOWLEDGEMENT. The author wishes to thank the referee for several vMuable suggestions.
REFERENCES
NJ/STAD, O.
4.
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