Download t-regular-closed convergence spaces

PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 51, Number 2, September
1975
T-REGULAR-CLOSED CONVERGENCE SPACES
D. C. KENT, G. D. RICHARDSON AND R. J. GAZIK
ABSTRACT.
It is known
compactification
space.
is almost
It is shown
T-regular-closed
that
that
a convergence
identical
a less
extensions
restricitive
with
space
to a completely
the
class
universal
which
has
tegular
a regular
topological
of convergence
property
of the
spaces
have
Stone-Cech
compactification.
1. Introduction.
space
has
tinuous
a Hausdorff
functions
a convergence
space
space.
In this
versal
property
sure
paper
spaces.
by relinquishing
Instead,
space
the filters
iff the space
having
the requirement
highly
that
topological
which
has
a uni-
of con-
nonidempotent
the
that
is regular
regular
compactification"
spaces
than,
for con-
of [6] and u] tot a class
we require
but more general
convergence
property
as a completely
a "regular
includes
Hausdorff
In [7] two of us showed
compactification
convergence
we obtain
(which
A convergence
between
a regular
that each
with an extension
regular
like the compactifications
be compact.
resembling,
has
ultrafilter
spaces
operators)
space
compactification
into compact
and has the same
vergence
In [6] one of us showed
clo-
"compactification"
that it be T-regular-closed,
a concept
compactness.
(X, —>) consists
of a set
on X and the elements
X and a relation
of X, subject
"_,"
to the following
conditions:
(1) x'-.
x, all x eX;
ra<f
.
..
(2) J —> x and , cr
J < (ú
g implies
,
, cr
.
(3) J* —»x implies
For
x e X, x'
empty
subset
expression
x".
denotes
g —» x;
S n x —»x.
the fixed
of X, then
"A —» x"
(o
cr
ultrafilter
A* will denote
should
be read
generated
the filter
"the
filter
by ixi; if A is a non-
of all oversets
.f converges
We will use the abbreviation
"u.f."
Throughout
will mean convergence
Presented
the paper,
to the Society,
space
for "ultrafilter".
June 21, 1975; received
AMSiMOS)subject classifications
54D35.
Key words and phrases.
Convergence
regular-closed
spaces,
T-regular-closed
of A. The
to the point
space.
by the editots
We will
May 25, 1974.
(1970). Primary 54A05, 54A20, 54D10, 54D25,
spaces,
extensions.
T-regular
convergence
spaces,
T-
Copyright © 1975, American Mathematical Society
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
461
462
D. C. KENT, G. D. RICHARDSON AND R. J. GAZIK
usually
refer
to a space
dorff if each
filter
to be Hausdorff
Fischer
unless
"cl,,"
definition
"X"
rather
is the closure
operator
standard.
cL v J —* x; XX, the topological
of the set
alent
X equipped
structure
space
regular,
if X is a topological
tion of important
is productive
space
set of any T-regular
space
in which
which
than
the
iff A = civ- A.
A T-
are equiv-
in the next
regularity,
sec-
T-regularity
spaces,
it can be embedded.
[4, Example
spaces
that
space
T-regua T-
a continuous
is closed.
in T-regular-closed
regular
sub-
of T-regular-closed
things,
of spaces
compact
A compact
3.10] describes
A study
other
classes
locally
if X is a closed
spaces.
are shown
spaces,
func§4 is
to be T-reg-
and lattices
with
convergence.
X be a space,
X with the coarsest
relative
f(x),
to which
C (X) the set
convergence
the natural
is continuous.
c-embedded
(R denotes
if the evaluation
of continuous
structure
map
(called
real-valued
continuous
line with its usual
on
convergence)
cú: C (X) x X —* R, defined
the real
functions
by &)(/, x) =
topology.)
X is
map /: X —» C C (X), defined by /(x)(/) = fix),
all / £ C(X), is an embedding.
spaces
Feldman
[2] has proved that all c-embedded
are T-regular.
A convergence
contains
Proof.
space
a compact
Proposition
hence
coarser
are given
onto a T-regular
T-regular
The following
c-embedded
Let
filter
on X.
it J—►
of regularity
T-regular-closed
among
space
with embedding
2. Examples.
order
J
of X, is the space
topology
Like
is not compact.
in §3; we show,
tion from a T-regular-closed
ular:
spaces.
T-regular-closed;
space
is given
concerned
modification
Some examples
will be called
is obviously
regular-closed
spaces
J" is a filter
A is ÀX-closed
of T-regular
J —» x whenever
and hereditary.
A T-regular
lar space
is Haus-
are assumed
X to be T-regular
and the two versions
space.
classes
for X and
with the finest
of X. A subset
is clearly
A space
All spaces
if cl^
We define
consisting
convergence
—>)".
X to be regular
x implies
regular
"(X,
one point.
indicated.
a space
has become
than
to at most
otherwise
[3] defined
—• x, where
This
as
converges
2.1.
Let
closed.
A locally
J —» x. Then
As a subspace
lows from [7, Lemma
clx J = cL^
X is said
J.
Since
to be locally
compact
if each
convergent
set.
l] that
compact
J
regular
contains
a set
space
A which
of X, A is a compact
cl^
and cl^x
X is regular,
cl,y
X is T-regular.
coincide
J —* x.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
regular
is compact
space,
for subsets
and
and it fol-
of A; thus
463
T-REGULAR-CLOSED CONVERGENCE SPACES
Pervin
gence
and Biesterfeldt
is regular.
T-regular.
[5] have
We will give
We use
the filter
ix 6 X, there exists
dually.
3
a lattice
with
order
showing
that
such
of order
convergence
conver-
a space
is
given
by
ordered set, J" a filter on X. Let L(J") =
F e 3 such that x < y fot all y e FÎ, and let 1/(3") be
order
converges
to x if x = inf Uij)
well known that order convergence
plete
that
proof
characterizations
Ward [9]. Let X be a partially
defined
shown
a shorter
= sup
need not be topological,
L(j ). It is
even in a com-
lattice.
Proposition
2.2.
Proof.
J
Let
Order convergence
order
converge
in any lattice
to x. Let
X is T-regular.
§ be the filter
generated
by all
sets of the form la, b] = {y: a < y < b\, fot a e L(5") and b £ i/(3).
Since
L(§) = L(j)
converges
to x. Also,
(see
and
t/(§)
sets
= (7(3),
of the form
[9J), which
is known
> y, and so cl^
investigated
on this
topic
closed
spaces)
topological
see
[lj.
space
spaces
Regular-closed
§ order
topology
convergence.
on X
Thus
3.1.
in various
Equivalently,
that
to that
space,
ways
spaces
for a summary
spaces
the concept
cl,y
J
(but not T-regular-
regular-closed
obtained
for topological
identical
to that
have
of results
of a regular-closed
of a topological
and the results
from those
3.1 is almost
for conver-
spaces.
of Theorem
2.10 of
be omitted.
A T-regular
J
topological
convergence
in [4]. Note
of Theorem
ter 3 072 X, cl.v
filter
that
in the interval
order
Regular-closed
convergence
differ
Theorem
than
of mathematicians;
is not equivalent
[4] and will therefore
closed
spaces.
are studied
The proof
la, b] ate closed
to be coarser
by a number
(or T-regular-closed)
gence
immediately
J —> x.
3. T-regular-closed
been
it follows
space
X is T-regular-closed
has an adherent
a T-regular
iff, for each
fil-
point.
space
X is T-regular-closed
iff each
maximal
space
is T-reg-
on X converges.
Corollary
3.2.
A closed
subspace
of a T-regular-closed
ular-closed.
Theorem
space
3.3.
X onto a T-regular
Proof.
Let
3 on ¡A such
/_1(3)
Let f be a continuous
VA',
space
A be a closed
that
where
3 —< y in
Y.
Then
subset
Y. Let
A* is the filter
function
from a T-regular-closed
f is a closed
of X, y e cly, /A.
map.
Then there
§ be an u.f. on X which
of all oversets
of A. Since
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
is an u.f.
is finer
X is
than
464
D. C. KENT, G. D. RICHARDSON AND R. J. GAZIK
T-regular-closed,
is closed,
there
A e cl.v-
is an adherent
a,
We next
there
set
3.4.
are
of the real
Let
and only
infinite
structure
such
subset
that
The spaces
However,
infinite
3A
and
no free u.f.
can converge,
Theorem
3.5.
set,
and
§^
which
// X:
the first
and
X
x £ Xy
maps
—»x in Xj.
Lemma
Proof.
such
map
P.
there
is a filter
Hence,
K = (P^
the second
ent point
Then
PjH
sub-
A such
that
B of X. Let Xj be
A ¿ —> x in Xj
subset
A.
by Theorem
in the product
then
that
3.1.
space
is a closed
g
E2^l
// X.
finer
clxx
Conversely,
J
has an adherent
map,
than
P.cl^y.
P.cl^y,
§) Vcl,y
projection
Xj
is T-
map
J
let
3 = cl^y
J
such
is a filter
P2 is closed,
J
point.
P$*
that
tj —> x,
on Y, and
cl^y
P2J\ =
such that A —>y, fot some y £ X,.
- ^> ana- so ^2^1 —' ^ *n *2'
is compact
X, x X.
3.3.
cl^y
K = Kj is a filter on Y. Also, P,H, > clxx
Similarly,
3.6.
infinite
are closed.
§, and so PjK
Hence> K. "~►
to cl xy 3.
regular
and
X2 is T-regular-closed,
then
is T-regular-closed.
It is easy
to see that the second
projection
Let J be a filter on Y = Xj x X2« By hypothesis,
y. Hence
g converges
contain
that
infinite
by Theorem
(x, y) in Y, and so (x, y) is adherent
the product
to each
T-regular-closed
show
PAli. Let K be an u.f. finer than P Ji
Then KVPJ
in X.
X with the finest convergence
are T-regular-closed,
is necessary
projection
cl^y. K = H. Since
point
of X (c the cardinality
which
the diagonal
Y = Xj x X2; we must
by hypothesis
that
and so Xj x X2 is not T-regular-closed.
The condition
on
for some
contains
A
of cl ^y, 3,
The example
subsets
be the set
X2 ate clearly
point
x a fixed
subset
structure
X
iff both projection
Proof.
be a filter
convergence
A. Let
since
is not productive.
of X, we can assign
C¡A —» x in X2 fot each
X.
regular-closed
and
infinite
on each
c subsets
free u.f.'s,
X with the finest
for each
Since
spaces.
T-regular-closed
X be a countable
2C free u.f.'s
line)
However,
is an adherent
of T-regular-closed
of being
[j.
y e/A.
B lt B 4 A and 3^ 4 §B for all infinite
the set
x X.
products
A of X two distinct
^A ^
3 —> y. Thus
that the property
Example
Since
cl.y,
consider
shows
x of cl^.
and so x e A. But /(x)
and so fix) = y, since
follows
point
there
is a filter
in X2 to y. Let
—♦ x, fot some
adherent point of cl.y
g finer that
clxx
clxx
H be an u.f. on V finer
x e Xj,
and
map
P,
P'23 = E2clXy
than
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
^> and
P~ !(§) VclAy,
P2K —i y in X2, so that
3-
is closed.
P23 has an adher-
(x, y) is an
3.
465
T-REGULAR-CLOSED CONVERGENCE SPACES
Theorem
closed,
3.7.
then
// both
X.
and
Proof.
It is sufficient,
jection
maps
in Y = Xj x X , and let
which
P.
B, and so
are known
and T-regular-
P,
point
is a closed
map.
topological
to be false,
The question
and Lemma
of whether
regular-closed
a product
is an unsolved
rem 3.5 is valid
P.
Then
and contains
to prove
that the pro-
is closed,
there
let
is an u.f.
a compact
set
B be
J
on
A. By Lemma
Also, B. = B Ci (Ax X A is a closed
an adherent
For regular-closed
that
P.S.
3) V B*. is such that
to [j, g has
P.cly,
compact
theorem,
To show
x 6 clx
to x in X,
dxx
preceding
are closed.
is T-regular-closed.
§ = (P~
belongs
P2
converges
3.6, A x X
Hence,
and
by the
closed
P.ß
X2 are locally
X. x X2 is T-regular-closed.
spaces,
§ = §, and since
Hence
x £ P.cly
Corollary
3.2 and Theorem
3.6 and Theorem
3.7 are known
topological
3.3
to be true.
spaces
We do not know whether
topological
A x X2
B. C
of regular-closed
problem.
for regular-closed
clxy
(x, b).
set.
is
or not Theo-
spaces.
A. Embedding theorems.
Theorem
closed
4.1.
space
Proof.
Each
T-regular
space
X can be embedded
in a T-regular-
X..
Assume
that
X is not T-regular-closed.
Let
y be a point
X, and let X, = X U iyl; let Xj be the set X U iyi, equipped
convergence
structure
satisfying
x 4 y, iff J
contains
X and the restriction
3 —» y iff 3 > y" A §, where
has no adherent
point
that X is a subspace
adherent
an adherent
ent to cl^x
3.
assumption
that
We shall
in which
erty.
each
Let
cluding
compactification
X
in X, then
by Theorem
to be the set
ing the following
cly, 3 —» x in X
A —» x in X.,
X.
for
to X in X;
X such that
that
cl^x
is T-regular,
has
the problem
of AX; let
Y, equipped
whenever
space
such
space.
guarantees
of finding
that
§
and
Let
with the finest
that
a class
extension
on X
y is adher-
Finally,
X is dense
the
in Xj.
spaces
with universal
map from
convergence
does
of T-regular
V = ßXX denote
(1) If 3 is a filter
3j
of X.,
AX is a completely
(f> be the embedding
<p
to X. If cl^x
is T-regular-closed.
a T-regular-closed
topological
conditions:
J»
by the construction
3.1, X.
X is not T-regular-closed
X be a T-regular
Hausdorff)
containing
to verify
3 has a restriction
now consider
member
conditions:
of 3 to X converges
of Xj. Let 3 be a filter on X., If 3 =y , then y is
point
Thus,
§ is a filter
in X. It is easy
to y. Otherwise,
not have
the following
not in
with the finest
prop-
regular
(in-
the Stone-Cech
AX into
structure
containing
Y. Define
satisfy-
(fix, then
3 —» tp~ x in X; (2) If § is an u.f. on X
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
466
D. C. KENT, G. D. RICHARDSON AND R. J. GAZIK
which
in
contains
X
cbX and
to the same
Theorem
then
X
4.2.
Z, then
to which
to AX-converge,
it converges
// X is a T-regular
space
and the ¡unction
a unique
First
tion of X
The first
on X
that
X
that
X
condition
is continuous;
is a filter
continuous
note
it is clear
injection.
l,[j>
fails
is a dense
f : X
is T-regular.
The function
of X
that
(pX and
regular
space
—> Z.
Y; from this
guarantees
regular,
embedding.
map into a compact
in the definition
cly, J, where (f>X£J
§ converges
Y.
is finer than
contains
cl„
and XX is completely
extension
the second
which
in
then
c/j: X —> X
if f: X —' Z is a continuous
f has
Proof.
—> X
point
is T-regular
Furthermore,
(p~ §
fact
and the construc-
(f> is certainly
guarantees
that
(fiX is dense
an
(b: X
in X . If Cj
g —» x in X , then,
by Condition
and (f>~ J —' (f>~ x in X. Since X is T-regu-
lar, clxx <p- l<3f— <p~lx in X. But 4>~*g > <p~!clAx. 3 = clxx (p—!3, and
so (f>~ § —• (p~ x in X. Thus
dense
(f>~
is continuous,
and
<p: X —» X
isa
embedding.
Finally,
Proposition
consider
/: X -* Z. Then
l], AZ is a compact
extension
f : Y --> AZ. Let
/: AX —►
AZ is continuous.
Hausdorff
topological
§ —» x in X . Assume
space.
first
By [7,
Thus
that
/ has
y > cly
an
J,
where tp"1? — (f>~Xxin X. Then /*(3) = fich~ *3) -» fi<p~ lx) = f*x in Z.
Also,
cl„
uous,
and so / (cly
/ 3 —' f x in Z, since
Z. Next,
assume
4>X such
that
that
g > cly
(p~ 3
fails
It follows
3, where
f : X
—» Z is continuous.
that
on X
/ g —* f x in
which
J —» x in
AZ have
By regularity
/§> AclY 3)> clz /*3, which implies
that
Then
l], Z and
and so / 3 —»/ x in Z.
But / : Y —» AZ is continimmediately
3 is an u.f.
to AX-converge.
—►/ x in AZ. By [7, Theorem
vergence,
Z is regular.
3) > cl2 / 3.
contains
Y, and so f J
the same
ultrafilter
of Z, clzf
con-
3—> f x, and
/*§ -*• /*x. Thus we have established
The uniqueness
of the extension
is obvious,
and so the proof is complete.
Unfortunately,
the conditions
enough
to insure
that
dition,
we obtain
the desired
Theorem
regular
4.3.
and locally
X
Let
compact.
continuous
has a continuous
closed
Proof.
By adding
T-regular-closed
Then
function
extension
In view of Theorems
Since
on X in Theorem
is T-regular-closed.
X be a T-regular
of X, and each
is T-regular-closed.
imposed
space
4.2 are not
an additional
such
that
XX is completely
(X , t/j) is a T-regular-closed
from
con-
extension.
X into a compact
regular
extension
space
Z
to X .
3.3 and 4.2, it remains
AX is locally
compact,
only to show that
AX is an open subspace
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
X
T-REGULAR-CLOSED CONVERGENCE SPACES
of Y = /SAX. Thus
no u.f. on
If g is an u.f. on X
z in
Y - (f>X. Thus
g > cly
suits
Y - cßX can converge
- tpX, then
there
//. From the definition
lead us to conclude
relative
to u.f.'s
X
contain
and
Y to a point
relative
K on (f>X such
of X , Condition
that
which
g converges
is an u.f.
in
467
to
that
Y to some point
K —» z in
at the conclusion
and so AX
is compact.
the same
Y — r/jX. Furthermore,
To show that
maximal
closed
the maximal
filters
X
closed
(relative
filters
to
Theorem
then
if g —> x in
described
the same
it is sufficient
X ) converges.
relative
to X
// X has
coincides
u.f. convergence,
to show that
By the preceding
are the same
are fixed,
a regular
since
every
paragraph,
as the maximal
Y is compact.
compactification
with the regular
From
[7, Theorem
and
Stone-Cech
l], X is T-regular
In the proof of the preceding
theorem,
the same convergence
for u.f.'s
account
the respective
sired
have
facts,
closed
Thus
X
XX is locally
compactification
com-
of X
in ll].
Proof.
1 ar.
AX
Y and
these
and the proof is complete.
4.4.
X
to
Y, and the latter
is T-regular-closed,
pact,
Y and
re-
u.f. convergence
AX = Y.
is T-regular-closed,
filter
relative
that
Thus,
Y, and
2, g —» z in X . These
Y have
4>X £ g, then (ß~ § —* x in AX, and so g —* x in AX . Combining
we arrive
in (pX.
and comparing
and
AX is completely
it is shown
containing
regu-
X
and
Y have
Y - rpX. By taking
this
fact into
constructions,
that
one
is
led
to the
de-
conclusion.
In [8, V3], a space
pact,
c-embedded,
However
AX
and have
is not locally
T-regular-closed.
X satisfying
the closure
operator
number.
blence
which
a completely
regular
compact,
hand,
a simple
in contrast
of a regular
to any topological
alteration
iterations,
in Theorem
com-
modification.
that
of this
X.
is not
example
4.3 with the property
where
to the stringent
compactification,
as described
to be locally
topological
of Theorem
for X has 72-distinct
extension
is shown
and it can be shown
the conditions
Therefore,
[7] for the existence
ular-closed
is constructed
On the other
to a space
natural
X
that
72 is an arbitrary
conditions
a space
leads
required
can have
4.3 and yet bear
little
in
a T-reg-
resem-
space.
REFERENCES
1. M. P. Berri, J. R. Porter and R. M. Stephenson,
Jr., A Survey of minimal topological
spaces,
General
Topology
and its Relations
to Modern Analysis
and Algebra,
III (Proc. Conf., Kanpur, 1968), Academia, Prague, 1971, pp. 93-114. MR 43 #3985.
2. W. A. Feldman, Axioms of countability
in C(X), Pacific
81-89.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
J. Math. 47 (1973),
468
D. C. KENT, G. D. RICHARDSON AND R. J. GAZIK
3. H. R. Fischer, Limesräume, Math. Ann. 137 (1959), 269-303.
4. D. C. Kent and G. D. Richardson,
Math. Soc» 160 (1971), 487-499.
5. W. J. Pervin
Minimal
convergence
MR 22 #225.
spaces,
Trans.
Amer.
and poséis,
Proc.
MR 44 #3279.
and H. J. Biesterfeldt,
Jr.,
Regular
topologies
Amer. Math. Soc. 17 (1966), 903-905. MR 34 #8368.
6. G. D. Richardson,
A Stone-Cech
compactification
for limit
spaces,
Proc.
Amer.
Math. Soc. 25 (1970), 403-404. MR 41 #992.
7. G. D. Richatdson
and D. C. Kent,
Regular
compactifications
of convergence
spaces, Proc. Amer. Math. Soc. 31 (1972), 571-573. MR 44 #3290.
8. -,
A note on C (X), Proc. Amer. Math. Soc. 49 (1975), 441-445.
9. A. J. Ward, On relations
between
certain
intrinsic
topologies
in partially
or-
dered sets, Proc. Cambridge Philos. Soc. 51 (1955), 254-261. MR 17, 67.
DEPARTMENT OF MATHEMATICS, WASHINGTON STATE UNIVERSITY, PULLMAN,
WASHINGTON99163
DEPARTMENT OF MATHEMATICS, EAST CAROLINA UNIVERSITY, GREENVILLE,
NORTH CAROLINA 27834
DEPARTMENT OF MATHEMATICS, ARKANSAS STATE UNIVERSITY, STATE UNIVERSITY, ARKANSAS72467
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use