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CONCERNING SEMI-CONTINUOUS FUNCTIONS
Dragan S. Jankovic
(received 22 April, 1981)
In 1963, N. Levine [4] introduced the concept of semi-open sets and
semi-continuous functions.
A subset
A
of a topological space
said to be semi-open if there exists an open set
U c A c cl(U) ,
f
function
cl(U)
where
U
X
X .
in
X .
A
V
Y ,
in
Y
f 1(7)
S. Crossley and S. Hildebrand [l] defined a set
to be semi-closed, if its complement is semi-open.
semi-closure of a set
U
into a topological space
is said to be semi-continuous if for any open set
is semi-open in
is
such that
denotes the closure of
from a topological space
X
A
They also defined the
in a topological space, denoted
be the intersection of all semi-closed sets that contain
sc04) ,
A .
to
In [2]
T. Hamlett investigated some questions concerning semi-continuous
functions into Hausdorff spaces.
The purpose of the present note is to
generalise and improve the following theorem in [2].
Theorem A.
Let
X
Hausdorff space,
(1)
(2)
If
f
be an arbitrary topological space and let
Let
f
and
g
be functions from
is continuous and
g
is semi-continuous, then
{x € X : f(x) = g(x)}
is semi-closed.
If
g
f
is continuous,
dense subset
D
of X ,
Throughout this note
and
f : X —y Y
X
is semi-continuous, and
then
and
f = g
Y
on
f
Our main result is the following theorem.
109
f = g
Y .
on a
X .
from a topological space
Y .
Math. Chronicle 12(1983) 109-111.
into
be a
will represent topological spaces
will denote a function
into a topological space
X
Y
Theorem.
of
If g : X — + Y
X x y,
then
is semi-continuous and
p^[S fl G(g))
S
is a closed subset
is semi-closed in
denotes the graph of a function
g
and
X ,
where
p^ : X * Y — * X
G{g)
represents the
projection.
Proof.
Let
X x Y ,
V
let
x € sclp^lS n G{g))) ,
let
U
S
where
is a closed subset of
x
be an arbitrary open set containing
be an arbitrary open set containing
g
is semi-continuous,
(10
is semi-open in
g{x)
X .
in
Y .
in
X ,
and
Since
g
It is shown in
Theorem 1.9 of [l] that an intersection of an open set with a semi-open
set is semi-open.
u € [u fl G 1 (7)) (1
g(u) € V .
that
U fl g 1 (10
Therefore
(5 (1 G(g)) .
Since
S
is closed,
and the result follows.
Corollary 1.
f
:
X
— *■
(u,g{u)) € S
This implies that
x € Pj (5 fl G{g))
If
Since
and
(U x V) fl S t <J> and, consequently,
Therefore
[x,g(x)) € cl{S) .
is semi-open.
[U n g'hv)) n P l ( s n G(g)) t * • Let
a e sc(P l (5 n <;(£))) ,
Y
(a:,<7 (x)) 6 S fl G{g) .
has a closed graph and
g
:
X
— *■
Hence
Y
is
semi-continuous} then
(1)
{x t X : /Or) = g(.x)}
(2)
If f = g
Proof.
on a dense subset
X
space
D
of X ,
then
f = g
{a: € X : f(x) = g{x)} = Pj [G(f) H G(g))
Since
is a closed subset of
Theorem.
is semi-closed.
X x y ,
on
X .
and since
£(/)
the first result follows from our
It is shown in Theorem 2.4 of [2 ] that the subset
is dense if and only if
D
of a
sc(D) = X . Therefore, the second
assertion follows from the first.
A function
every
x € X
f : X — >■ Y
is said to be weakly-continuous [3] if for
and every open set
V
containing
110
fix)
in
Y
there
exists an open set
U
containing
x
in
X
such that
/(£/) £ cl(V) .
It is shown in Theorem 10 of [5] that a weakly-continuous function into
a Hausdorff space has a closed graph.
Corollary 2.
If f : X — ► Y
continuous, and
Y
Therefore, by Corollary 1 follows
is weakly-continuous,
g : X — *■ Y
is semi-
is Hausdorff, then
(1)
{x € X : f(_x) = g{x)}
(2)
If f = g
is semi-closed.
on a dense subset
D
of X ,
then
f = g
on
X .
Finally, the fact that continuity implies weak-continuity gives that
the Theorem A is the consequence of Corollary 2.
REFERENCES
1.
S.G. Crossley and S.K. Hildebrand, Semi-closure, Texas J. Sci. 22
(1971), 99-112.
2.
T.R. Hamlett, Semi-continuous functions, Math. Chronicle 4(1976),
101-107.
3.
N. Levine, A decomposition of continuity in topological spaces,
Amer. Math. Monthly 68(1961), 44-46.
4.
N. Levine, Semi-open sets and semi-continuity in topological
spaces, Amer. Math. Monthly 70(1963), 36-41.
5.
T. Noiri, Between Continuity and Weak-continuity, Boll. U. M. I.
(4) 9(1974), 647-654.
University of Belgrade
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