I nternat. J. Math. & Math. Sci.
Vol. 6 No.
197
(1983) 197-199
A NOTE ON ALMOST CONTINUOUS MAPPINGS
AND BAIRE SPACES
JING CHENG TONG
Department of Mathematics
Wayne State University
Detroit, Michigan 48202
(Received April 20, 1982 and in revised form June 4, 1982)
ABSTRACT.
We prove the following theorem:
THEOREM.
Let Y be a second countable, infinite
countably many open sets
01, 02,
in Y such that
0n,
If there are
R0-space.
0...,
01 02
then a topological space X is a Baire space if and only if every mapping f:
It is an improvement of a theorem due
is almost continuous on a dense subset of X.
to Lin and Lin
[2].
KEY WORDS AND PHRASES.
Sparaon axiom
RO,
almost contuous mapping, Baire space.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES.
54D10.
1.
X- Y
Primary 54C10, 54F65; Secondary
INTRODUCTION.
This note is directed to mathematical specialists or non-specialists familiar
with general topology [i].
Lin and Lin
[2] proved the following theorem:
THEOREM I.
Let Y be an arbitrary infinite Hausdorff space.
logical space such that every mapping f:
X
If X is a topo-
Y is almost continuous on a dense
subset D(f) of X, then X is a Baire space.
In the theorem above, the almost continuity is in the sense of Husain [3].
The proof of the theorem depends on the following lemma (cf. Long [i,
Prob. 14,
p. 1471):
LEMMA I.
subspace.
Every infinite Hausdorff space contains a countably infinite discrete
J. TONG
198
In this note, we prove a lemma similar to Lemma 1 under weaker conditions,
and use it to improve Theorem i.
PRELIMINARIES AND RESULTS.
2.
Before stating the result, we first recall the definition of the separation
axiom R
0
(cf. [4], [5], [6, p. 49]).
A topological space X is R
DEFINITION 1.
open subset U, x
0
U implies {x} e U.
It is known [i] that R
0
T
T
I
O
+ R0.
LEMMA 2.
sets
is weaker than T
If an infinite space X is
such that
n
countably infinite distinct set S
01 be
01 02 we
E
R0,
in fact
and there are countably infinite open
On
{yl,Y2,
n
Yl
TO,
and is independent of
01 02
n, there is an open set Vn satisfying V
PROOF.
1
A Hausdorff space is R
0.
O
01, 02
if and only if for each x e X and
Yn’
then there is a
"’’} in X such
{yn }.
n S
Without loss of generality we may assume that
an arbitrary point.
Since X is
that for each
R0, {yl}
c
01
is not empty.
Let V
01.
I
0
Let
From
I.
02n(Ol\{Yl}).
n-27--Then V is an open set and
chosen and Vn_l=On_l n (0n_2\Ui__l{Yis)
2
Y2 V2" If
O n, we may choose
is defined, then since On_
1
Yn On such that Yn 0n-i and
{yn 0n. Let Vn 0n n (On- --i ). Then Yn Vn. Thus we have a countably
can find a
Y2
infinite distinct set S
VI,V2
(On_ Ui= 1
Yn+m 0n+m-I
sets
Vn,
we have
but 0
n
E
02
such that
{yl,Y2,
such that
{y2
02.
Let V
Y2 Ol
Yn-i is
and
Yn
and countably infinite distinct open
Yn E Vn
Yi Vn for i 1
0n+m_ 1 hence Yn+m
(n
2
).
1,2
n-l.
Since
On’ Yn+m
V
n
Since
2
Vn 0n n
Yn+m On+m
Therefore
(m > i)
{yn
V nS
n
For convenience we say that a space X has an ascending chain of open sets if
there are countably infinite open sets
01,02
On
such that
01 $ 02 %...
On
LEMMA 3.
An infinite Hausdorff space X is an
R0-spade
with an ascending chain
of open sets.
PROOF.
We need only to show that X has an ascending chain of open sets.
Lemma i, there is a countably infinite discrete subspace
{yl,y 2
’Yn
},
By
hence
A NOTE ON ALMOST CONTINUOUS MAPPINGS AND BAIRE SPACES
UI,U 2, .,Un,
01,02,...,On,... is an
there are disjoint open sets
(n
1,2,...).
Then
such that
Yn E Un.
Let
199
In
0n
Ui__ Ui
ascending chain of open sets.
The converse of Lemma 3 is not true.
EXAMPLE i.
Let X
X is R and 0
X\{ i__i
i
0
X is not Hausdorff.
X\N; N
[0,i] with topology
1
...}(i
"i+l
1,2
is a countable
set}.
Then
is an ascending chain of open sets.
Now Theorem 1 can be imp roved as
THEOREM 2.
Let Y be an infinite
R0-space
with an ascending chain of open sets.
If X is a topological space such that every mapping f:
on a dense subset of
X, then X
X
Y is almost continuous
is a Baire space.
The proof is all the same as the proof of Theorem 2 in [2].
Similar to Theorem 3 in
THEOREM 3.
we have
Let Y be a second countable infinite
chain of open sets.
every mapping f:
[2],
X
Ro-space
with an ascending
Then a topological space X is a Baire space if and only if
Y is almost continuous on a dense subset of X.
REFERENCES
i.
LONG, P.E. An Introduction
Columbus, Ohio, 1971.
2.
LIN, S-Y.T. and LIN, Y-F.T.
3.
HUSAIN, T.
4.
DAVIS, A.S. Indexed system of neighborhoods for general topological spaces,
Amer. Math. Monthly 68 (1961), 886-893.
5.
SHANIN, S.A.
(1943)
On separation in topological space, Dokl. Akad. Nauk. USSR. 38
110-113.
6.
WILANSKY, A.
Topology for Analysis, Ginn, Waltham, Mass., 1970.
to General Topology, Charles
E. Merrill Publ. Co.,
On almost continuous mappings and Baire spaces,
Canad. Math. Bull. 21 (1978), 183-186.
Almost continuous mappings, Prace Math. i0 (1966), 1-7.