Download H-CLOSED SPACES AND THE ASSOCIATED 9

H-CLOSED SPACES AND THE ASSOCIATED 9-CONVERGENCE SPACE
T.R. Hamlett*
(received 23 May, 1978; revised 25 September, 1978)
The concept of 9-convergence was introduced by N. Velicko [8 ], and
has also been studied under the name "r-convergence" by L.L. Herrington
and P.E. Long [4].
In this paper characterizations of quasi //-closed,
C-compact, //-closed, and //-closed Urysohn spaces are obtained in terms
of natural conditions on the associated 9-convergence space.
Product
theorems for quasi //-closed and //-closed spaces are also given.
For a given topological space
A
interior of
T-nbd
by
cl 04)
(X, T ) , we denote the closure and
and int(4)
respectively.
(neighbourhood) system at a point
9 -nbd system at
U e N(x)} .
for every
filterbase
V e N q (x )
x , denoted
A filterbase
V e N^(x)
F
x e X
N a (x) , is defined as
F e F
is said to 9 -accumulate to
of all ordered pairs
N(x) , and the
N a (x) = {cl (i/):
is said to 9 -converge to
F
there exists an
there exists an
by
We denote the
F e F
x
such that
{ (F,x) : F-^x, x e X}
gence space or structure associated with
x (f — ► x) , if
0
such that F £. V . A
[4], if for every
F fl V t <$> .
The system
is called the 9-conver­
(X , T ) .
The reader is
referred to [6 ] for the definitions of 9-cluster point, ^-closure3
and 9 -continuous maps, but we remark that these are merely the closure
operator and continuous maps associated with the 9-convergence space.
Note that
(Z,9(!T))
is a limit space in the most restricted sense
of Fischer [3] (principal and pretopological are both terms which are
* Author partially supported by a grant from Arkansas Tech. University.
Math. Chronicle 8(1979) 83-88.
83
T\
used in the literature to refer to such spaces), and is a
gence space in the sense of Ordman [7].
conver­
We will use [7] as a reference
for the general theory of convergence spaces.
Definition.
Q-closed [ 8 ] if
A .
A
A
A subset
of a topological space
A = clg04) , where
is said to be B-open if
cl^CA)
X - A
(X,T)
is said to be
denotes the Q-closure of
Q-closed.
is
The collection of 0-open sets form the topology associated with the
convergence
Tq
where
9(T)
T^ .
and is denoted by
T
is the semi-regularization of
topological space
(X,T )
[l].
A subset
is said to be an H-subset [8 ] of
{U
for every collection
— ^s
We remark that
: a e A}
A
3
of a
iX,T)
if
of open sets such that
A c u {U :a 6 A}, there exists a finite subcollection such that
—
a
A c U{cl(t/ ) : i=l,2,...,n} . X is said to be quasi E-closed if it
i
is an //-subset of itself. A quasi //-closed Hausdorff space is called
an H-closed space.
topological space
Note that these definitions are for an arbitrary
(X , T )
with no separation properties assumed.
Also
note that a subset of a topological space can be an //-subset and not
be quasi //-closed as a subspace (see the example on p. 106 of [ 8 ]).
If
A
is a subset of a topological space
will denote the elements
F
A .
is a filterbase on
general the same as
induced on
A
by
(F,x)
of
Q(T)
(X,T ) , then
such that
x e A
Note, however, that
6 (^4 ) » where
T'
and
is not in
denotes the subspace topology
T .
A convergence space is said to be compact if every maximal filterbase converges.
Lemma.
Let
(A,Q(T)a )
(XST)
be a topological space with
A
x .
is compact if and only if each filterbase on
Then
A
has a
Q-accumulation point.
The proof is straightforward and hence omitted.
theorem is the main result of this paper.
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The following
A subset
Theorem 1.
if and only if
of a topological space
(A3 Q(T)^)
Necessity.
Proof.
A
F
on
A
is an H-subset
is compact.
We suppose that
proceed to a contradiction.
base
(X3T)
(i4,0(T)^)
is not compact and
By the Lemma, there must exist a filter-
that does not 9-accumulate to any
a e A .
Thus for
a e A , there exists a U c N(a) and an F e F such that
’
a
a
F fl cl (U ) = 6 . Now there exists by assumption a finite subcollection
a
a
each
{U
: i = 1,2, ... ,n}
a.
of
i=l,2,...,n} .
’’ ’
F
Now
F
F
o
Let
± <i> implies
r
{U \ a z A]
a
e F
o
F
o
such that
F
such that
fl cl (U
a.
4 £ U{cl (£/ ) :
a.
c f1{F
: £=1,2,...,n) .
o —
a.
t
) ? <f> for some 1 5 i 5 n . Hence
J
n cl(U 1 t <f> , and this is a contradiction.
a.
a .
J
J
Sufficiency.
We show the contrapositive using the Lemma.
exists an open cover
U = {U
a
: a e A}
A
of
If there
such that
A - U{cl(£/ ) : i= l,2,...,n} t
for all finite subcollections of
ai
U , then define F to be the collection of all sets of the form
A - U{cl(£/ ) : i=l,2,...,n}
F is a filterbase on A that does
a.
%
not 0-accumulate to any point in A , and the proof is complete.
Corollary 2.
A topological space
if
is compact.
(X3 Q(T))
Corollary 3.
(X3 T q )
(XST)
If a topological space
is quasi H-closed if and only
(X3T)
is quasi H-closed3 then
is compact.
The proof follows from corollary 2 and Theorem 3.16 of [7].
Since
in general not every closed subset of a quasi //-closed space is quasi
//-closed, we have the following definition due to G. Viglino [9]: A
topological space is C-compact if each closed subset of the space is
an //-subset.
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Corollary 4.
(A3 Q(T)^)
A topological space
(XST)
is C-compact if and only if
is compact for every closed subset
A
of
(X>T) .
The characterization for ff-closed spaces is given in the following
theorem.
A convergence space is said to be
T2
more than one limit point, and is said to be
an element
{a;}
converges to any point
Theorem 5.
A topological space
Urysohn) space if and only if
Proof.
Observe that
(X,T)
(Xt Q(T))
(X,9(T))
is
if no filterbase has
Tj
if no filterbase with
y ? x .
is an H-closed (H-closed
is compact
T\{T 2 )
T^(T2) .
if and only if
(X,T)
is
Hausdorff (Urysohn), and apply Corollary 2.
Theorem 6.
The Q-continuous image o f a quasi H-closed space is quasi
H-closed.
Proof.
The proof follows from Corollary 2 and the fact that the
continuous image of a compact convergence space is compact.
To conclude the paper we use the above theorem to give easy proofs
of the following two product theorems.
Theorem 7.
A nonempty product o f quasi H-closed spaces is quasi
H-closed if and only if each factor is quasi H-closed.
Proof.
Since necessity is an immediate consequence of Theorem 6 , we
need only show sufficiency.
Let
{(Z^,? ) : ot e A)
of nonempty quasi #-closed topological spaces.
{(^, 0 ( 2 ^ ) )
spaces.
: a e A)
is a collection of nonempty compact convergence
Theorem 3.18 of [7] (Tychonoff's Theorem) then implies that
(ttX^ f r O ^ ) )
is compact.
For a point
form
denote a collection
Then, by Corollary 2,
(a^) e
, it is easy to see that all sets of the
tt{cl (Z/ ) : £=l, 2 ,...,n} * v{Xn : 3 ^ a.}
a.
3
t
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constitute the nbd
system at
(x^)
with respect to both
Consequently, we have that
space.
Therefore
(ttZ ^ ttT^)
©(irT^)
(ttX^, ©(ttT^) )
and
ttG
^)
.
is a compact convergence
is quasi //-closed by Corollary 2, and
the proof is complete.
Theorem 8.
A nonempty product of H-closed spaces is H-closed if and
only if each factor is H-closed.
Proof.
The proof follows readily from the previous theorem and
Theorem 1.3 of [2].
REFERENCES
1.
M.P. Berri, J.R. Porter, and R.M. Stephenson, Jr., A survey o f
minimal topological spaces3 General Topology and its Relations
to Modern Analysis and Algebra, 111 (Proc. Conf. Kanpur, 1968),
Academia, Prague, 1971, 93-114.
2.
J. Dugundji, Topology} Allyn and Bacon, Boston 1966.
3.
H.R. Fischer, Limesraume, Math. Ann. 137(1959), 269-303.
4.
L.L. Herrington and Paul E. Long, Characterizations o f C-compact
spaces j
5.
Proc. Amer. Math. Soc. 52(1975), 417-426.
Paul E. Long, An Introduction to General Topology}
Charles E. Merrill Publishing Co., Columbus, 1971.
6.
Takashi Noiri, Properties o f 0 -continuous functions, Atti.
Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Natur. (8) 58(1975),
No. 6, 887-891.
7.
Edward T. Ordman, Convergence and Abstract spaces in Functional
Analysis (Part 1)3 Journal of Undergraduate Mathematics, Sept.,
(1969), 79-95.
87
8.
N. Velicko, H-closed. topological spaces3 Amer. Math. Soc.
Tra n s l . 78(2) (1969), 103-118.
9.
G. Viglino, C-compact spaces3 Duke Math. J. 36(1969),
761-764.
Arkansas Tech. University
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