Download A. X s-oonverges to a point p € X (denoted by F

CHARACTERIZATIONS OF REGULAR-CLOSED SPACES
Larry L. Herrington*
.
(received 11 March, 1975; revised 21 April, 1976)
1.
Introduction
Characterizations of regular-closed spaces are given in terms of
open regular filterbases [l, p.104].
Open filterbases, of course,
determine nets but not every net determines an open filterbase.
In
this paper we give characterizations of regular-closed spaces in terms
of nets and arbitrary filterbases.
These characterizations are
obtained mainly through the introduction of a type of convergence for
filterbases and nets that we call s-convergence.
Throughout, cl04) will denote the closure of a set
A
will denote the set of accumulation points of a filterbase
will assume that the condition of regularity includes the
and
4(F)
F.
We
T^
separa­
tion axiom.
2.
Preliminary definitions and theorems
Let
A
be a non-empty subset of a topological space
S(A) = {0^ : A c 0^,, a € A}
Then
S(A)
X
and let
be a family of open subsets containing A.
is called a shrinkable family of open sets containing
if for each
0
a
€ S(A) , there exists an
On € S(A)
8
A
such that
cl(O a) c 0g .
Definition 2.1.
A filterbase
X s-oonverges to a point
• a ^ A}
p € X (denoted by F
in a topological space
p) if for each
*This research was supported by a grant from the National Science
Foundation, Grant No. SER76-08651, Research Initiation for Minority
Institution Improvement.
Math. Chronicle 5(1977) 168-178.
168
shrinkable family S(p) = {0
there exists an
0o € S(p)
F s-accumulates
to
an
0o i Sip)
8
p
X
ct
€ F
8
F
a
p,
c 0o.
p
p) if for each shrinkable
±<fs for each
a
F
such that
s
of open sets containing
0 o fl F
such that
F
(denoted by
: 3 € A}
p
of open sets containing
and some
p
S(p) = {0
family
: 6 € A>
p
p, there exists
a ( A ,
Convergence and accumulation of filterbases in the usual sense,
of course, imply s-convergence and s-accumulation, respectively.
How­
ever, the converses do not hold as the next example shows.
Let
Example 2.2.
I = [0,1] have as a subbase the usual open sets
together with the set
F - {F
n
Z = { r : 0 < r < l
: n £ N} (where
irrational}) and let
F = {a; : -rr < x <
n
2 n +1
2
p = h • Then
late in the usual sense to
to the point
r
and
p
but
F
F
is rational}.
n+1
x
and
Let
is
does not converge or accumu­
s-converges and s-accumulates
p.
There are a number of theorems concerning s-convergence and
s-accumulation of filterbases whose statements parallel those of
convergence and accumulation of filterbases in the usual sense.
We
state a few of these theorems but omit their straightforward proofs.
In a topological space
Theorem 2.3.
(a)
If
is a filterbase in
F
then
(b)
Let
Fj
than
Fj.
and
F2
Then
the following properties hold :
X such that
F s-accumulates to p.
converges to p, then
X
F
If X
s~converges to p € X,
is regular and if F
F s-accumulates at no point other than p.
be two filterbases on
F 1 s-accumulates to
X where
pi X
if
F2 is stronger
F2 s-accumulates
to p.
(c)
A maximal filterbase
(d)
only if M s-converges to p.
If X is a T^ normal space, then a filterbase
M
(accumulates) to p € X
in
X
s-accumulates to p t X
if and only if
169
if and
F converges
F s-converges
(s-accumulates) to p.
Definition 2.4.
Let
X.
be a net in
J
Then
be a topological space and let
s-converges
0
if for each shrinkable family
Q(T^) c 0
to
p $X
a
(denoted by
£ S(p)
c
If
: a € A)
p)
of open sets
0^ £ S(p) and some
d £D
0
such that
s-accumulates
p) if for each shrinkable family
s
of open sets containing p, there exists an
0
0 fl Q(T J t 0
a
a
such that
0: D — *■X
(denoted by 0
= {o £ D : d < c}) . The net
(where
S(p) = {0^ : a € A}
0
S(p) = {0
p, there exists an
containing
to p € X
0: D — * X
X, then the family
is a net in
F(0) = {0 (T^) : d € D}
d 6 D.
for each
X
is a filterbase on
and it is routine to
verify that :
(a)
F(0) — *■p € X
s
(b)
F(0) « p € X
if and only if
if and only if
S
Conversely, every filterbase
0: D — * X
F
0 — *■p. :
>
0 * p.
5
in
X
determines a net
such that :
p € X
(a)
0
(b)
0 « p £ X
if and only if
if and only if
p.
F
F « p.
The construction of such a net is the same as that of [3, p.213].
In a topological space
Theorem 2.5.
(a)
If
0
is a net in
X
s-accumulates to p.
X
the following properties hold:
that s-converges to p £ X3 then
If X
converges to p € X, then
0
is a regular space and if 0
0
s-accumulates at no point other
than p.
(b)
A universal net
0
s-accumulates to p £ X
if and only if 0
s-converges to p.
3.
Filterbase and net characterizations of regular-closed spaces
An open filterbase
F
in
X
170
is a regular filterbase if for
U € F, there exists a
each
Let
space
A
A = {Z7CX : a £
J.
and if
Then
8
S
a)
such that cl(V) c £/ [l].
F ( F
and
3 = {7Q
: 8 £ I} be covers of a
p
is a regular refinement of
if
is a shrinkable refinement of itself.
shrinkable refinement of itself if for each
€ 8
A
V
8
(8
refines
is a
€ B, there exists a
P
An open cover is regular if it
such that cl(V^) c y ).
has an open regular refinement.
Lemma 3.1.
Let
A[F) = y4g (F)
of
F be an open regular filterbase on
(where ^ S (F)
X.
Then
denotes the set of s-accumulation points
F).
Proof.
^ S (F) c A{F).
We only need to show that
p j: j4(F].
suppose that
containing
p
and some
Let
p € X
Then there exists an open set
U i F
such that
and
G(p)
U (] G(p) = 0 .
Then
S(p) = {X-cl(V) : V € F and cl(V) c [/} forms a shrinkable family of
open sets containing
p
and has the property that for each
(X-d(V)) € S(p )j (X-cl(V)) fl V = 0.
conclude that
Consequently, p j: A o (F).
We
-4(F) = A [F].
We next characterize regular-closed spaces in terms of arbitrary
filterbases.
Theorem 3.2.
In a regular space
(b)
X is regular-closed.
Each filterbase in X
(c)
Each maximal filterbase in
(a)
Proof,
X
X
the following are equivalent:
s-accumulates to some point p € X.
X
s-converges to some point p £ X.
(a) implies (b). Suppose there exists a filterbase,
that has no s-accumulation point.
• exists a shrinkable family
S(x)
Then for each
x € X3 there
of open sets containing
171
~ 0
x
such
’
It follows that collection of open sets A = {U(x) : U(x) € S(x)s x t X}
that for each
U(x) € S(x)s U(x) fl
F, in
f°r son,‘
e Fy(x) *
X.
forms a regular open cover of
Consequently, by Theorem 4 of
[l, p.104], there exists a finite subcollection,
6 = {U.(x.) Z A : i = 1,2,3,..., m
3
T'
property that
m
n(i)
U
U
^
j
Since
8
. = 0.
Choose
covers
contradiction.
FQ € F
such that
X, there exists a
(b) implies (a).
Let
F
X
n(i)
U (x7) fl F., , , j- 0
p k
Vp'xiJ
A (F)
s
which is a
0..
be an open regular filterbase on
* By Lemma 3.1 and hypothesis (b) we have that
Therefore
m
c fl
fl F .
i=l J = 1 j i
* 8 with the property
Therefore,
We conclude that
U-(x .) € 8
t
i
F n D U (x,) i 0.
U p k
that
Now for each
FU .(x.) £ F such that
3 ^
there exists a corresponding
.
j = l,2,3,... ,n(i) }, with the
U.(x.) = X.
3 t
i=1 j=l
U.(xJ fl F
and
X.
A(F) = A (F) j^0.
s
is regular-closed according to Theorem 4.14 of
[1, p.104].
(b)
M
implies (a).
Let
s-accumulates to some point
M
be a maximal filterbase in
p tX
and hence s-converges to
X.
Then
p
by Theorem 2.3 (c).
(o)
implies (b).
Let
F
exists a maximal filterbase, M, in
s-converges to some point
be a filterbase in
X
stronger than
F.
p £ X, F s-accumulates to p
X.
Then there
Since
according
to Theorem 2.3.
Since filterbases and nets are "equivalent" in the sense of
s-convergence and s-accumulation, we can now characterize regularclosed spaces in terms of nets.
Theorem 3.3.
In a regular space
X
(a)
X
(b)
Each net in
(c)
Each universal net s-converges.
the following are equivalent:
is regular-alosed.
X
has an s-aocumulation point.
172
M
If a regular space
Lemma 3.4.
filterbase in
Suppose that
point.
Fix
p
that
has the property that every
X with a unique s-accumulation point is convergent
then every filterbase in
Proof.
X
p £X
F
X
has an s-accumulation point.
X
is a filterbase in
and define
with no s-accumulation
= {F U {p} : F ( F}.
is the unique s-accumulation point of
converge to
F^
and
It follows
F^
does not
p.
If S(A) and S(B) are shrinkable families containi-ag
Lemma 3.5.
A and B respectivelys then S(A U B) = {U U V : U £ S(A) and V € S(B)}
is a shrinkable family containing
Proof.
A U B.
The straightforward proof is omitted.
Remark 3.6.
Let
F
be a filterbase on
X
A c X.
shrinkable family on open sets containing
S(A)
and let
be a
We say that
F
misses S(A) if for each U € S(A) , there exists a corresponding
Fy £ F with the property that cl (U) fl F^ = 0.
A
Suppose that
p j: A.
is a closed subset in a regular space
Furthermore, assume that
F
is a filterbase on
misses a shrinkable family
S(A)
that contains
exists a shrinkable family
S q (A)
containing
p f cl(U)
has the property that
let
V(p)
define
p
that
Then there
that misses
U € S q (A).
such that
and
F
and
To see this,
A fl V(p) = 0
and
S 0(A) = {U fl (X-zl(H(p)) : U £ S(A), Hip) is open
and p £ H(p) c cl (H(p)) c V(p)}.
It follows that
contains
for each
be an open set containing
A
A.
X
X
S q (A)
A, misses
is a shrinkable family of open sets that
F, and has the property that
p £ cl[i/ fl (X-cl(H(p)))] for each [jj fl (X-cKH(p)))] £ S Q(A).
In
view of the preceding discussion, we now give the following definition.
Definition 3.7.
Let
F
be a filterbase on a regular space
173
X,
p € X, and let
C = (SiA) : S(A)
is a shrinkable family of open sets containing
A c X-{p} }.
a closed set
We define
S = {S(A) t C : S(A) misses F and p ^ cl (U(A)) for each U(A) € S(A)}.
Theorem 2 of [2, p.454] characterizes minimal regular spaces in
terms of open regular filterbases.
In our next theorem we character­
ize minimal regular spaces in terms of arbitrary filterbases.
Theorem 3.8.
A regular space (X3i) is minimal regular if and only if
eaoh filterbase in
Proof.
X with a unique s-accumulation point is convergent.
Suppose the condition is given and let
filterbase on
* 3.1, p
X
with a unique accumulation point
p.
converges to
p ( I.
F.
is the unique s-accumulation point of
hypothesis, F
F be an open regular
This shows that
By Lemma
Consequently, by
X
is minimal
regular according to Theorem 2 of [2, p.454].
Conversely, assume that
suppose that
point
p
X
is a minimal regular space and
F is a filterbase on
X
with a unique s-accumulation
to which it does not converge.
Let
S
be the set given
in Definition 3.7 and define
S = {H c X-{p} : H
(Xjj)}
is open in
U {(X-cl(U(A))) U G(p) : U(A) € SiA), S(A) * S, G(p) € t, and
p € G(p)}.
{H c X-{p} : H € t)
It is clear that the collection
for a topology on
X-{p}.
Now let
(X-cKUfA^)) U G(p)
be elements in
B
forms a base
and
(X-cl (U(AZ))) U Hip)
containing the point
p.
We note that
p € [X-cl (U(Al) U U(A2))] U (G(p) fl Hip))
C
[(X-cliUiA^)) U Gip)]
n
[ a - c l (U(AZ))) U Hip)].
But according to Lemma 3.5 and Remark 3.6,
174
[X-cliUiA^ U U(A2)j ] U (G(p) fl H(p)) t B .
Consequently, it follows that the collection
a base for a topology
8
xQ
X
on
with
t
0
c
xQ
x.
containing
S(x) € 5
shrinkable family
J(x) € tq
We next show that
Kix) c X-{p} be
Let
F « x3 there exists a
Since
x.
that contains
Then there exists an open set
of open sets forms
.
t
X.
forms a base for a regular topology on
an open set in
8
U(x) € S(x).
Let
x
containing
with the
property that
a: € J(x) c cl ^iJix)) c \_K(X) fl U(x)~\ c K(x)
c !q (J(x ))
(where
(We note that
Thus, Tq
J(x)
denotes the closure of
p ^ cl Q(J(x))
iX3x^)).
J(x) D (X-cl(U(x))) =0.)
since
is regular at each point
in
x £ X-{p}.
Now let
(X-cKU(A))) U H(p)
p.
be a basic open set containing
G(p)
exists an open set
€
(X3\)
Since
containing
t
p
is regular, there
such that
p $ G(p) c cl (G(p)) c H(p).
(We note that
clQ (G(p)) = cl (G(p)).)
Now since
A, there exists a
shrinkable family containing
S(A) 6 S
V(A) € S(A)
is a
with
the property that
cl (U(A)) c V(A) c c l(V(A)).
Consequently,
p € [X-cl(V(A))] U G(p)
c cl0[ a - c lfVfAjJJ U C-fpJ]
c [X-cl(U(A)j] U H(p)
showing that
tq
p.
is regular at
open set
U(p)
F i F .
But for each basic open set
€
t
containing
Pj there exists a corresponding
[j-cl(7J(A))~\ 3 Fu .
that
T0 i x
Thus
showing that
p
By hypothesis, there exists an
such that
F
U(p)
for each
[Z-cl (I)(A)) ] U Hip) containing
F^ £ F with the property that
[X-cl(U(A))] U Hip) <f. Uip). We conclude
(X3t) is not minimal regular.
175
4.
First countable regular spaces.
(X3x)
A space
if
t
is called first countable and minimal regular
is first countable and regular, and if no first countable
X
topology on
which is strictly weaker than
is first countable and regular-closed if
(Xs t )
regular, and
t
x
is regular.
(Xs\)
is first countable and
is a closed subspace of every first countable
regular space in which it can be embedded [4 ].
A first countable regular space
Theorem 4.1.
X
is first countable
and regular-closed if each countable filterbase on
X
s-accumulates
to some point p £ X.
Let
Proof.
F
X.
be a countable open regular filterbase on
By Lemma 3.1, A[¥) = ^4^{F ) / 0
X
which implies that
is first
countable and regular-closed according to Theorem 2.6 of [4, p.116].
A first countable regular space
Theorem 4.2.
and regular-closed if each sequence in
X
X
is first countable
s-accumulates to some
point p £ X.
Suppose that
Proof.
X
is not regular-closed.
Y
a first countable regular space
h : X — * h(X) c y
exists a sequence,
p £ Y-h(X).
Since
is not closed in
f : N — + h(X) s in
h(X)
h(X)
is homeomorphic to
z € h(X). Therefore
some point
and a homeomorphism
h(X)
such that
Then there exists
z = p
Y.
Thus there
converging to some point
X, f s-accumulates to
according to Theorem 2.5 (a)
which is a contradiction.
Theorem 2.6 of [4, p.116] shows that a first countable regular
space
X
is first countable and minimal regular if every countable
open regular filterbase on
convergent.
X
with a unique accumulation point is
In our final theorem we show that a space
countable and minimal regular if each sequence in
176
X
X
is first
with a unique
s-accumulation point is convergent.
Lemma 4.3.
If a regular space
sequence in
Suppose that
tion point.
n
X
has an s-accumulation point.
(x^)
p £X
Fix
is odd and
z^
is a sequence in
X with no s-accumula­
and define a sequence (z
if
(z^)
s-accumulation point of
Theorem 4.4.
has the property that every
X with a unique s-accumulation point is convergent
then every sequence in
Proof.
X
n
is even.
Then
p
by
z^ = p
if
is the unique
to which it does not converge.
A first countable regular space
(X>x)
countable and minimal regular if every sequence in
is first
X with a unique
s-accumulation point is convergent.
Proof.
Suppose that
mapping of
(X,x)
h 1 is continuous.)
converging to a point
point of the sequence
that
h(p)
is a continuous bijective
onto a first countable regular space
need to show that
Y
h: (X,x) — *■(Yyo)
y € Y.
(y )
Let
If p 6 X
Y
be a sequence in
(h~^(y ))s then the continuity of
is regular, h(p) = y.
(We
is an s-accumulation
is an s-accumulation point of the sequence
Therefore, since
(Yso).
h
shows
(y ).
Consequently, h l(y)
the unique s-accumulation point of the sequence
hypothesis, the sequence (h~l(y )) converges to
-1
n
that h
is continuous.
(h ^(y^)).
h l(y)
By
showing
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M.P. Berri, J.R. Porter and R.M. Stephenson, A Survey of
Minimal Topological Spacess General Topology and Its Relations
to Modern Analysis and Algebra, Proceedings of the Kanpur
Topological Conference (1968), 93^114.
2.
M.P. Berri and R.H. Sorgenfrey, Minimal Regular Spaces, Proc.
177
is
Amer. Math. Soc, 14(1963), 454-458.
3.
J. Dugundji, Topology, Allyn and Bacon, Boston, Mass. 1966.
4.
R.H. Stephenson, Jr., Minimal First Countable Topologies>
Trans. Amer. Math. Soc. 138(1969), 115-127.
University of Arkansas at Pine Bluff