Download A NOTE ON SEMITOPOLOGICAL PROPERTIES D. Sivaraj

A NOTE ON SEMITOPOLOGICAL PROPERTIES
D. Sivaraj
(received 11 May 1982, revised 16 November 1982)
Introduction
1.
A
Let (*,T) be a topological space and
A
and interior of
A
is said to be regular open if
A = C(KA)).
RO{X,t)
subset
A
exists
G € t
of
X
A = l(C(A))
G c A <- C(G).
such that
SO(X,t).
X, and
A
where
whose closures cover
SO{Xtt) = SO(X, ta) .
X.
A subset
A
(X,t), if (i4,t|i4) is S-closed where
is S-closed relative to
has a finite
X
of
A
A* - l(C(_I(A))).
of
t |i4
X
X
X.
xa
is
is a
has a finite subfamily
is an S-closed subspace of
is the relative topology on
subfamily whose closures cover
x i X
If
t°
A topological space (;£,t) is
t, [9], if every cover of
locally S-closed, [10], if each
an S-closed subspace of
in a space (^,t) is
t c t° c 50(^,t),
S-closed, [16], if every semi-open cover of
X
and regular closed if
The family of all semi-open sets
A subset
A £ A*
said to be an a-set, [8], if
of
The closure
respectively.
in a space (^,t) is said to be semi-open, [7], if there
the family of all a-sets in (X,t), then
A
X.
1(A)
and
is the family of all regular open sets in (*,t).
in (X,t ) is denoted by
topology on
a subset of
C(A)
in (.£,t) are denoted by
A
A.
A.
by semi-open sets
A space (*,t) is
has an open neighbourhood which is
A space (^T,t) is quasi-H-closed (QHC), [ll],
(feebly compact [l]) if every open (countable open) cover of
X.
finite subfamily whose closures cover
X
has a
A space (£,t) is nearly compact,
[l3] (mildly compact, [l4]) if every regular open (countable regular open)
cover of
X
[15], if each point of
(*,t).
X
is an intersection of regular closed sets of
A space (X, t ) is extremally disconnected if the closure of every
A space (X,t) is almost regular, [1 2 ] if for each point
open set is open.
x € X
A space (X,t) is weakly Hausdorff,
has a finite subcover.
and each regular closed set
joint open sets
U
and
V
A
such that
Math. Chronicle 13(1984) 73-78
73
not containing
x € U
and
x, there exist dis­
A <- V.
A space (X,t) is
pseudocompact, [18] , if and only if every continuous real-valued function on
X
is bounded.
X
If
X .
If
T(X)
is a set of points, let
t € T(X)
and
is the equivalence class of topologies on X
[t]
which yield the same semi-open sets as
of
[t] , denoted by
be the lattice of topologies on
F(t) [4] .
(I,t) , then there is a finest element
A property of topological spaces is defined
to be a semi-topological property, [5] , if it is preserved by semi-homeomorphisms (bijections so that the images of semi-open sets are semi-open and
the inverse images of semi-open sets are semi-open).
The following lemmas
will be used in the sequel.
Lemma 1.1
If
[5].
f : (X,x) — *• (Y,o)
f : (XyF(t) ) — ► (Y, F(a) )
Lemma 1.2 [10].
then
I(C(A))
In a space
each
x € X,
(X,\),
if
A space
(X3t)
is S-olosed relative to
there exists
t,
t.
is locally S-closed if and only if for
V € t
such that
x € V
and
V
is S-closed
x.
Lemma 1.4 [lO].
If
X
is a locally S-closed space and
open and continuous surjection, then
2.
A
is also S-olosed relative to
Lemma 1.3 [10].
relative to
is a semi-homeomorphism3 then
is a homeomorphim.
Y
f : X~*-Y
is an
is locally S-closed.
The Results
The following first two lemmas play the key role in this paper.
Lemma 2.1.
Proof.
t“ = F(t).
The result follows from proposition 4 of [8] and theorem 2 of [2].
Lemma 2.2.
C*(A)
For a space (X,t)3
In a space (X,t), if
is the closure of
A
A € SO(X3 t ) ,
then
C*(A) = C(A)
with respect to the topology
74
F(i).
where
Proof.
if
Since
x c F(t) ,
x t C*(A),
A c
V €
F(t)
there exists
which implies that
and so
for any
104) fl I(K) =
x,
C*{A) c C(A).
such that
A c £7(J(j4))
neighbourhood of
x,
and so,
x t C(A) .
x € V
and
C(J(i4)) fl 7* = <j>.
A fl V* = <p.
Therefore
Since
A fl V »
I(j4) fl C(I(V)) = <p
which implies that
1(A) fl V* - <f>, which implies that
semi-open,
For the conyerse,
V*
Since
4
is
is a t-open
C(A) c £*(>4),
which proves the
lemma.
Lemma 2.3.
A space (X,x) is QHC (feebly compact) if and only if (XsF(t))
is QHC (feebly compact).
Proof.
If (X.FCt)) is QHC, by lemma 2.2, (Jf,x) is QHC.
(X,x) is QHC, since for each
[X,F(t)) is QHC by lemma 2.2.
V t F(x),
V c V*,
V* € t
Conversely, if
and
(7(7) = C(V*),
The proof for feebly compact spaces is
similar.
Theorem 2.1.
The property of a space being QHC (feebly compact) is a
semi-topological property.
Proof.
Since the homeomorphic image of a QHC (feebly compact) space is a
QHC (feebly compact) space, proof follows from lemmas 2.3, and 1.1.
Since
RO(X,t) = ^ ( ^ ^ ( t ) ),
by proposition 6 of [8], we have the
following
Lemma 2.4.
A space (X3x) is nearly compact (mildly compacts weakly
Hausdorff) if and only if
(X3F(x))
is nearly compact (mildly compact,
weakly Hausdorff).
Theorem 2.2.
The property of a space being nearly compact (mildly compact>
weakly Hausdorff) is a semi-topological property.
Proof.
Since the homeomorphic image of a nearly compact (mildly compact,
weakly Hausdorff) space is a nearly compact (mildly compact, weakly Hausdorff)
space, proof follows from lemmas 2.4, and 1.1.
75
Theorem 2.3.
The property of a space being extremally disconnected is a
semi-topological property.
Proof.
By proposition 7 of £8], (*,x) is extremally disconnected if and
only if
(Z,F(x))
is extremally disconnected.
Since the homeomorphic
image of an extremally disconnected space is an extremally disconnected
space, the proof follows from lemma 1 .1 .
Lemma 2.5.
A space (X3t) is almost regular if and only if
(X,F(t))
is
almost regular.
Proof.
suppose
If (X,x) is almost regular, there is nothing to prove.
(Z,F(t))
is almost regular.
U
there exist disjoint sets
U*
Then,
and
and
V*
and
If
A
Conversely,
is regular closed and
V € F(x)
such that
x € U
and
are the required disjoint open sets such that
x f. A,
A c v.
x € U*
A c V * . Hence (X,x) is almost regular.
Theorem 2.4.
The property of a space being almost regular is a semi-
topological property.
Proof.
Since the homeomorphic image of an almost regular space is an almost
regular space, the proof follows from lemmas 2.5, and 1.1.
Lemma 2.6.
x
In a space (Xjt), a subset
if and only if
Proof.
Since
Lemma 2.7.
A
A
of
X
is S-closed relative to
SO(X,x) = 50(*,2:’(x)),
is S-closed relative to
F(i).
the proof follows from lemma 2.2.
A space (X3t) is locally S-closed if and only if
(XyF(r))
is
locally S-closed.
Proof.
Suppose
(Jf,f(T))
lemma 1.3, there exists
relative to
lemma 1.2,
F(x).
J(C(7))
is locally 5-closed.
V € F(x)
By lemma 2.6,
such that
V
x i V
x € X.
and
V
is 5-closed relative to
is 5-closed relative to
(*,x) is locally 5-closed.
Let
x.
Since
Then, by
is 5-closed
x.
By
x € V c /(CCF)),
Conversely, if (X,x) is locally 5-closed,
76
since
t c F(t) , by lemma 2.6,
Theorem 2.5.
(^,F(t))
is locally
S-closed.
The property of a space being locally S-closed is a semi-
topological property.
Proof.
The proof follows from lemmas 2.7, and 1.4.
Theorem 2.6.
The property of a space being pseudocompact is a semi-
topological property.
Proof.
If
C(X3t)
is the set of all continuous real-valued functions on
(Z,t) , since the real line with the usual topology is regular, by proposition
8 of [8] ,C{X, t) = C(XtF(t))
and so,
(£,F(t))
Then, since the homeomorphic image of a pseudo­
is pseudocompact.
is pseudocompact if and only if
compact space is a pseudocompact space, the proof follows from lemma 1 .1 .
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