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AMERICANMATHEMATICAL SOCIETY
Volume 178, April 1973
MONOTONICALLYNORMALSPACES
BY
R. W. HEATH(l), D. J. LUTZERÍ1), AND P. L. ZENOR
ABSTRACT.
property
cept
This
of linearly
of monotone
for stratifiability
paper
normality
to certain
1. Introduction.
to give
concludes
to give
results
ordered
which
The
con-
conditions
theorem
of a metrization
for p-spaces
theorem
relate
due to
monotone
properties.
paper
normality
Heath
spaces—but
to Zenor's
In [29],
normality
([17],
original
holds
without
[5].
monotone
and Lutzer
unexpectedly
fails
appears,
spaces
relating
complementary
normality
first
Orz stratifiable
results
Subsequently
that monotone
linearly
proof
a common
spaces.
and sufficient
of examples
and announced
[7] announced
ing, in particular,
necessary
of monotone
and stratifiability.
of spaces—the
a list
topological
The property
normality,
stratifiable
a new metrization
an easy
with
familiar
the property
to metrizability
and Borges
is used
2.1 of C. R. Borges'
named
of monotone
and of Borges'
and to provide
The
in Lemma
the study
spaces
of a T. -space,
Treybig.
P. Zenor
begins
normality
with G g-diagonals,
name,
paper
ordered
[18])
work,
in one large
to hold in others
where
showclass
it might
be expected.
This
paper
and by Heath
of Lemma
authors
is a combination
and Lutzer.
2.2 which,
would
The set of natural
2.1.
which
to each
conducted
T
and
for suggesting
results.
"mapping"
is denoted
A space
ordered
X is
pair
this
Throughout
means
monotonically
and Lutzer.
The
format.
this paper
"continuous
by the letter
by Zenor
with the exception
§§5 through 7, is due to Heath
the editor
numbers
Definition.
assigns
open set
with
and preliminary
to be at least
studies
§§2 through 4 are the work of Zenor,
together
like to thank
2. Definitions
assumed
of the independent
all spaces
are
onto function."
zV.
normal
ÍH, K) of disjoint
if there
closed
is a function
subsets
G
of X an
G(/7, K) such that
(a) H C GÍH, K) C G(W, K)~ C x\K;
(b) if (H , K ) is a pair of disjoint
Received
by the editors
October
AMS(MOS)subject classifications
54E35, 54F05.
Key words and phrases.
spaces,
extension
theorems,
theorems,
Monotone
p-spaces,
closed sets having
15, 1971 and, in revised
H C H and K D K
form, May 25, 1972.
(1970). Primary 54D15, 54D20; Secondary 54E20,
normality,
collectionwise
normality,
stratifiable
Gg-diagonal,
linearly
ordered spaces,
metrization
ßN.
0) Partially supported by NSF Grant GP 29401.
481
Copyright © 1973, American Mathematical Society
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482
R. W. HEATH, D. J. LUTZER, AND P. L. ZENOR
then GÍH, K) C G(ß', K').
The function
G is called
2.2.
Any monotonically
Lemma.
ator G satisfying
sets.
(a)
each
of a space
There
arated^)
normality
normal
space
operator
fot
X.
has a monotone
normality
oper-
GÍA, B) DGÍB, A) = 0 for any pair (A, B) of disjoint closed
Furthermore,
normality
a monotone
of the following
is a function
subsets
properties
is equivalent
to monotone
X:
G which
of X an open set
assigns
to each
ordered
pair
(S, T) of sep-
G(S, T) satisfying
(i) S C GÍS, T) C GÍS, T)~ C x\T;
(ii)
if (S , T ) is a pair of separated
sets
having
S C S and
T D T' then
GÍS, T)CGÍS', T').
(b) There
closed
and
is a function
p £ X\C,
H which
an open set
assigns
to each
ordered
pair
(p, C), with
C
Hip, C) satisfying:
(i) p £ H(p, C) C X\C;
(ii) if D is closed and p i C D D then Hip, C) C Hip, D);
(iii) if p / q are points of X then Hip, \q\) C\ Hiq, \p\) = 0.
Proof.
To prove
the first
ator for X and define
statement,
let
G be any monotone
G(/4, B) = G\A, B)\G'iB,
normality
oper-
A) . Then G is a monotone
normality operator for X having GÍA, B) CtGiB, A) = 0.
Clearly
any pair
(a)=>(monotone
normality)=»(b).
ÍS, T) of separated
sets
We show that
(b) implies
(a).
For
let
GiS, T)=\J\HÍp, T~)\ pe S].
Clearly S C GÍS, T); and GÍS, T)~ C X\T since for each q £ T, the set Hiq, S")
is a neighborhood
of q disjoint
from G(S, T). Property
(ii)
of (a) follows
directly
from (ii) of (b).
2.3. Remarks,
complete
monotone
(a) The property
normality
(b) The equivalence
independently
monotone
2.4. Definition.
5: [closed
(a) with monotone
[7] who has given
called
several
other
normality
properties
was obtained
equivalent
to
in [8].
(c) Lemma 2.2 makes
tion
in 2.2 (a) was originally
[29].
of statement
by Borges
normality
described
it clear
A space
subsets
that monotone
(X, 7) is
normality
semistratifiable
of X| x N —►J, called
is a hereditary
[12] if there
a semistratification
property.
is a func-
of X, such
that:
(a) If H C K are closed subsets
of X then S(H, n) C SÍK, n) for each n £ N;
(b) H =C\\S(H, n)\ n £ N\ tot each closed set H C X.
(2) Two subsets
H, K of X are separated
ii HC\K~ =0=KDH~.
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483
MONOTONICALLY NORMAL SPACES
If, in addition,
the function
S satisfies
(c) H = C\[SÍH, n)~\ n e N\ for each closed set H C X,
then
X is
stratifiable
2.5. Theorem.
A space
and monotonically
Proof.
fication
Then
[5] and
S is called
a stratification
X is stratifiable
of X.
if and only if X is semistratifiable
normal.
Suppose
X admits
a monotone
T. For each closed
S is a stratification
set
normality
operator
G and a semistrati-
K C X and n e N let S(K, n) = GiK, x\t(K,
of X. The converse
follows
from a result
n)).
of Borges
[5, Lemma 2.1] and Definition 2.4.
2.6.
normal,
Theorem.
so is
Proof.
subsets
Suppose
f: X —» Y is a closed
mapping.
If X is monotonically
Y.
Let
G be a monotone
normality
operator
for X. For disjoint
closed
H, K of Y, let
G'iH, K) = y\/[X\G(/-HH), f~HK))]..
Then
G is a monotone
It follows
immediately
(or even hereditarily
normal
subspaces,
whether
is
7.1])
by a collection
properties
normal
taken
3.1.
wise
that
to be a discrete
Theorem.
of such
collection
provided
by a locally
of closed
It might be interesting
that
finite
monotonically
X is dominated
to know
(cf.
[5,
subspaces.
of monotonically
normal spaces.
discrete
collection
Recall
collection(4)
that a space
K of closed
X = iD(W)| H e K! of open subsets
H C D(H) for each
collection
y.
normal.
normal
[3] if for each
is a disjoint
with the property
(A)
X is monotonically
3. Special
of X there
for
that if X can be covered
preserving
be monotonically
collectionwise
sets
operator
from (2.6)
closure
then
X must
Definition
normality
H e M. It is easily
seen
that
X
subof X
5) may be
of open sets.
Any monotonically
normal
space
is (hereditarily)
collection-
normal.
Proof.
Let
G be a monotone
normality
operator
for X which
satisfies
GÍA, B) n GÍB, A) = 0 for each pair (A, B) of disjoint closed sets in X (cf.
Lemma 2.2).
Let K be a discrete
family
of closed
subsets
of X. For each
fi eH
define DÍH) = G{H, \J[H' £ K| H' / H\). If Hx ¿ H2 are members of K then
D(z7j) C GiHy H2) and D(H2) C G(Hr Hx) so that D(H x) O D(H2) C G(H , H2) n
G(H2, H x) = 0. Therefore 2) = [DiH)\ H £ K\ is the required disjoint collection
(3) A collection
Cl^uir.l
which
[S-\ i £ l\ of subsets
of X is hereditarily
i el\) = U{CIX(7\)| 1 e f| whenever T.CS.
(4) A collection
meets at most
H of subsets
of X is discrete
one member of K.
closure
preserving
if
for each i e I.
if each
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point
of X has
a neighborhood
484
R. W. HEATH, D. J. LUTZER, AND P. L. ZENOR
of open sets.
reditary
Thus any monotonically
collectionwise
3.2. Notation.
functions
from
3.3.
space
normality
For any space
Theorem.
Suppose
there
space
is collectionwise
He-
the set of all continuous
/ = [0, l].
H is a closed
is a function
normal.
from (2.3) (c).
Y, C(V, /) denotes
Y to the unit interval
X. Then
normal
follows
subspace
Q: C(ß,
of a monotonically
/) —> C(X, /) such
normal
that:
(a) for each f £ C(H, I), Qif) extends f, i.e., Qif)ix) = /(x) for each x £ H;
(b) if f, g £ C(H, I) and fix) < g(x) for every x £ H, then Q(/)(x) < Q(g)(x)
for each x £ X.
Proof.
The construction
D be a monotone
normality
number of the form p/2q,
open set Dfip/2q,
is of the type used to prove Urysohn's
operator
where
ip + l)/29)
for X. For each
Lemma.
/ £ CÍH, I) and for each real
- 1 < p < 2q and q > 0 are integers,
we obtain
such that:
(1) /" H[0, p/2q]) C Dfip/2q, ip + l)/2") C D/p/2«, ip + l)/2«)_ C
X\f'liiip + D/2«, 1]);
(2) DfHp - l)/2«, p/2qY C Dfip/2", ip + l)/2*);
(3) if p is odd, then DfHp - l)/2«, ip + l)/2«)~ C Dfip/2q, ip + l)/2«);
(4) if p is even, then Dfip/2q, ip + l)/2«)" C Dfip/2q, ip + 2)/29); and
(5) if /(x)<g(x)
for each
x in H, then
Dfip/2q,
ip + l)/29)
C
D ip/2q, ip + l)/2q) fot each p/2q.
To this
end, let / denote
a member of C(H, /). For all
Dfi-l/2q, 0) = 0
q > 0, let
and Dfi1, Í2q + l)/2«) = X.
Let
Hfio, y2)= /- Hloi),
Kfio, y2)= /- HEM,iJ)
and
D/0, ^) = D(H/0, M), K/0, '/0);
uß,
i) = /-1([o,H])uD/(o, JO",
k/1^, i) = /-1(U!)
and
z^Ci 1)= D(///(M,D.K/H, D)In general
Let
let
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an
1973]
MONOTONICALLY NORMAL SPACES
JDÍÍn
Hkn/2«, in + l)/2«) = f ^O. */2"]) u
485
l)/2<*, in + l)/2«r
{Díín - l)/2«,
[ÍD((t2
if n is odd,
n/2q)
ií 72 is even,
+ l)/2«,Gi + 3)/29) if 72is odd,"|
[Din/24. ín + 2)/2«)
if z2is evenl
and
0,01/2«, (n + l)/2«) = DÍHfín/2q, ín + l)/2«), K/rz/2«, (« + l)/2«)).
Now, to obtain
the function
Q, let / e C(W, /). For each
x in X, define
Qif)ix) = glbip/2«: x e D^p - l)/2«, p/2«)}.
It follows
from (1) that for each
that if for each
remains
x in H, fix)
to show
Before
that
Qif)
proceeding,
< gix),
note that it follows
x e X and let
= fix).
then for each
is in C(X, /); that
(6) if q > t and Dkp/2q,
(5 + l)/2'>(p
+ l)/2<".
Now, let
x in H, Qif)ix)
ip-l)/2q.
Qif)
from (2), (3), and
«7 be an integer
from (5)
< g(g)(x).
It
is continuous.
(4) that
x\Dfis/2l,
such
p be the fitst integer such that x is in D((p - l)/29, p/2q).
that ß(/)(x)>
it is clear
x in X, Qif)ix)
is, that each
(p + l)/2«) intersects
e > 0. Let
Also,
(s + l)/2'),
that
l/29
then
< e/2. Let
It follows from (6)
Let
1/ = D((p - l)^*,
p/2«)\cl
D((2(p- 1) -l)/29+1,
U is an open set in X that contains
x. Let
(2(p - l))/2(q +1)).
y e U. Clearly,
Qif)iy)
< p/2q
<
Qif)ix) + e. It follows from (6) that Q(f)iy) > (2(p - l) -l)/2(*+1); and so,
Q(/)(x) - í < p/21- - 1/2« - l/2(« +1>= (2(p + 1) - l)/2 (q +1)< Qif)iy). Thus,
lß(/)(y)-e(/)(*>l
<«.
Remarks,
(a)
(b) In Theorem
stratifiable
space
uous real-valued
formation
authors
normal
The authors
4.3 of [5],
do not know whether
Borges
proved
X and if C(W) and
functions
on H and
Q: CÍH) —*C(X) which
do now know whether
C(X)
are the real vector
X respectively,
satisfies
Borges'
the converse
that if H is a closed
then
(a) and
result
of 3.3 is true,
subspace
spaces
there
of contin-
is a linear
(b) of our Theorem
can be proved
of a
trans-
3.3. The
for monotonically
spaces.
4. Products
techniques
of monotonically
and results
4.1. Theorem.
of this
normal
section
spaces.
with those
If X x Y is monotonically
set of X bas a limit point
or Y
The reader
should
compare
the
of [19].
normal
then either
is stratifiable.
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no countable
sub-
486
R- W. HEATH, D. J. LUTZER, AND P. L. ZENOR
Proof.
Suppose
limit point
that
M = \m | n £ N\ is a countable
p. We may assume
monotonically
normal,
tone normality
operator
fined for pairs
of separated
sets
that
p i M . Let
so is the subspace
M x Y (cf.
for M x Y as described
subsets
subset
of X having
M = M u \p}. Since
2.3(c)).
in 2.2(a).
of M x Y.) For each
Let
G be a mono-
(Thus
G(S, T)
closed
a
X x Y is
set
is de-
F C Y, the
HF = |(x, y) £ M x Y\ y £ F and x / p} and Kp = |(p, y) e M x Y\ y i F\
are separated
GÍHp,
subsets
KA\.
4.2.
of M x Y. For each
It is easily
Corollary.
seen
that
re £ N let
T(F,
T is a stratification
The following
properties
re) = |y £ Y\ (zzz , y) e
for
of a space
Y.
X are equivalent:
(a) X is stratifiable;
(b) X x M is monotonically
(c) // M. = |0|
tonically
u |l/w|
, the product
Proof. Clearly
its usual
metric
space
topology,
M;
then
X x zVL is mono-
i.e.
so that
product
(li(re))
The definition
4.4.
then
X
and
normal.
normal.
a sequence
converging
to a
X x MQ is a subspace
of X x X
Thus
(c) =» (a)
(d) => (c).
is a nonstratifiable
of X, is monotonically
A completely
of covers
That
normal
regular
of X by open
space
for each
space
subsets
X such
X", the
n £ N.
X is a p-space
of ßX
that
such
if there
that
fllSt(x,
is a
(J(re))|
x £ X.(5)
of p-spaces
t^A-spaces
Corollary.
normal
contains
a copy of MQ. Then
that there
[l].
re e A/| C X for each
tonically
of X, is monotonically
from Theorem 4.1.
7.6 shows
of re copies
of M-spaces
many copies
X x MQ is monotonically
4.3. Definition
sequence
two points,
X contains
follows directly
Example
of countably
(a) => (b) =» (c) and (a) => (d). We show that (d) =» (c) =>
If X has at least
limit point,
= X
re £ N\ has
for every
normal;
(à) X
(a).
normal
will be needed
can be found
A space
and
in [25] and
X is metrizable
X satisfies
in a later
section.
The definitions
[6] respectively.
if and only if X2 = X x X is mono-
one of the following
conditions:
(a) X is a p-space;
(b) X is an M-space;
(c)
X is a w/S-space.
Proof.
satisfies
If X is discrete
then
X is metrizable.
(a), (b) or (c), then some countable
cording
to (4.1),
p-space
is a Moore space
a iM-space
X is stratifiable.
(see
and it was proved
(5) If H is a collection
It is shown
also
[9]); thus
If X has a limit point
subset
of X has a limit point.
Ac-
in [12] that a semistratifiable
X is metrizable.
in [6] that a stratifiable
of subsets
and
zM-space
Any M-space
is
is metrizable.
of X and if p e X, then St(p, II) = u\U e ll| peí/}.
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MONOTONICALLY NORMAL SPACES
V
Recall
subset
that a completely
of ßX
[13]
regular
and that
a metrizable
only if it is Cech-complete
4.5.
Corollary.
complete
and
4.6.
point.
Cech-complete
is completely
if X is a Ggmetrizable
if and
then
metrizable
countably
compact
space
such
that
X
= X x
X is metrizable.
then some countable
X is stratifiable.
if and only if X is Lech
normal.
// X is locally
normal,
If X is not discrete
Thus
space
X is completely
X x X is monotonically
Corollary.
X is
[10].
A space
X is monotonically
Proof.
space
Since
a stratifiable
subset
of X has a limit
countably
compact
space
is
V
compact
([14],
[12]),
X is locally
compact;
hence
X is Cech-complete.
Thus
X
is metrizable.
5. Monotone
normality
5.1. Definition.
Let
in ordered
(X, <) be a linearly
vex if [x e X| a < x < b] is a subset
be any subset
of X. A convex
mal with respect
Clearly,
components
5.2.
a triple
usual
any well-ordering
ordering
closed
tains
and
that
p £ X\C,
defines
let
expressed
of C. Let
CC S which
S
is maxi-
as a union of its convex
of a set are disjoint.
topological
ordering
is monotonically
X admits
of the set
which
C C X is con-
space
of the set
(abbreviated
LOTS)
X and where
3 is the
is
of <.
Any LOTS
We verify
A subset
C is convex."
components
ordered
< is a linear
topology
Theorem.
Proof.
linear
convex
(X, i, <) where
5.3.
"C C S and
A linearly
set.
a, b are points
of S is a subset
of X can be uniquely
and distinct
open-interval
ordered
of C whenever
component
to the property
any subset
Definition.
spaces.
lip,
a function
X. (The order
the topology
normal.
Gip, C) as in (2.2) (b).
W need
not have
any relation
of X.) For any pair
C) be the convex
component
Let
W be
to the
ip, C) where
of X\C
which
C is
contains
p.
If l_(p, C)= \y € lip, C)\y < p\ ¿0
let x(p, C) be the W-first point of I Jp, C)
and let yip, C) be the W-first point of I+ip, C) = jy 6 lip, C)\ y > p\ if /+(p, C)/ 0. Now define
\]xip, C), yip, C)[ if I_(p, C)/0/
|[p,y(p,C)[
Gip, C) =
]xip,C),p]
\p\
In any case,
Gip, C) is an open subset
the three conditions
l+ip, C);
if /_(p, c) = 0//+(p, C);
if I_íp, O¿0¿
l¿p,C);
if /_(p,C) = 0 = /+(p,C).
of X and it is clear
of (2.2)(b).
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that
Gip, C) satisfies
488
R. W. HEATH, D. J. LUTZER, AND P. L. ZENOR
Combining
(5.3),
5.4.
Corollary
5.5.
Definition.
is a topology
(2.3)(c)
[27]-
and (3.1) we obtain
Any
LOTS
is hereditarily
A generalized
on X and
ordered
< is a linear
(a) §i<) C J where
an easy
§i<) denotes
(b) each point of X has a local
is a triple
of X such
the usual
of the following:
collectionwise
space
ordering
proof
(X, J , <) where
J
that
open-interval
J-neighborhood
normal.
base
topology
of <;
consisting
of convex
sets.
It is known that a space
and only if there
is a linear
X can be topologically
ordering
of X making
embedded
in some
X a generalized
LOTS
ordered
if
space
([10], [20]). See Examples 7.1, 7.2, and 7.6.
5.6.
Corollary.
5.7.
Questions,
Any generalized
(a) There
ordered
are several
space
is monotonically
well-known
proofs
that
normal ([4], [22], [27]), all of which invoke the axiom of choice.
asks
whether
one can prove
As far as the authors
LOTS
which
know,
that any
that
[22] (another
property
Let us conclude
due to Treybig
5.8.
question
are order-complete
(b) What is the relation
is normal
is still
any
LOTS
In [4], Birkhoff
without the axiom
open (except,
is
of choice.
of course,
for
and order-bounded).
between
of any
this
LOTS
normal.
monotone
LOTS
section
which
normality
implies
with a simplified
and
K„-full
collectionwise
proof
normality
normality)?
of a metrization
theorem
[28].
Corollary.
Let
X x Y is a continuous
X and
image
Y be infinite
of a compact
Hausdorff
LOTS
L.
spaces
Then
and suppose
X and
that
Y are metriz-
able.
Proof.
a closed
Let
map.
f: L —►X x Y be continuous
According
X x V is compact;
limit point.
According
and compact,
X and
6. Metrization
6.1.
to (2.6),
thus
X and
to (4.1),
and onto.
Since
X x Y is monotonically
Y each
both
contain
X and
L is compact,
normal.
a countably
infinite
Y are stratifiable.
/ is
Furthermore
set with a
Being
stratifiable
Y ate metrizable.
of p-spaces.
Definition.
A space
X has a G^-diagonal
if the set
A„ = |(x, x): x e X}
is a Gg-subset of X x X.
It is known
of open covers
In proving
that
X has a Gg-diagonal
of X such that
our metrization
the other obtained
(6) See footnote
HlStOc,
theorem
if and only if there
is a sequence
§(re)): re £ N\ = |x} for each
we use
two results,
by Creede.
(5), p. 486.
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(y(w))
x £ X.(6)
one due to Bing
and
489
MONOTONICALLY NORMAL SPACES
6.2.
Theorem,
lectionwise
X such
(a)
that,
for each
(b) a completely
space
(cf. 4.3)
regular
space
of x such
semistratifiable
6.3.
that
p-space
Theorem.
p-space
if (x
base
covers
of
at x) [3];
if and only
is a sequence
z's a sequence
if X is col-
of open
if X is a p-
(g (x)) of open
in X and if y e g ix ) for
X is developable
if and only if X is a
1.2].
X is metrizable
if and only if X is a monotonically
with a G^-diagonal.
In the following
According
normal.
proof all closures
as described
Because
in Lemma
To accomplish
to Theorem
this
of ßX
normal
6.2(a),
such
H is
of
is collec-
to show that
(b) of Theorem
there
that
space
it suffices
we use statement
with a Gg-diagonal,
of X by open subsets
in ßX.
and if p jt q are points
3.1, a monotonically
according
X is a p-space
are taken
2.2(b)
of Hip, [q\).
to Theorem
Therefore,
X is developable.
covers
)
[12, Theorem
to be a function
Proof.
(§(rz))
n £ N\ is a local
x e X there
X we shall write Hip, q) instead
tionwise
if and only
is a sequence
to y, i.e.,
A space
Conventions.
assumed
X is metrizable
X is developable
point
n, then (x ) converges
normal
space
(= there
x e X, |St (x, §(«)):
and for each
neighborhoods
each
A regular
normal and developable
is a sequence
UÍn + 1) refines
6.2.
(iKrz))
liin)
of
for each
72 £ N and such that if x £ X, then HiSt (x, 11(m)): n £ N\ = [x\. For each y e X
and
n e N, let
Dniy)
D (y) be an open subset
is a subset
Suppose
of some
show that (x ) converges
(xn), no term of which
x
/ y, there
*-*».(*
";
nj
member
(x ) is a sequence
to y, it suffices
were
that
such
an integer
•»'-'*
(i) y e D (y) and
g (y) = Híy,
X\D
n e N. To
to show that any subsequence
that
cluster
point.
if / > /',, then
j > j,
such
that
ix
))CHix
x
Tiy
x
Fix
(ii)
(y)).
y £ g ix ) fot each
y as its unique
j,
that
of ll(zz). Define
in X such
is y, has
is an integer
.)• If there
of X such
(x
) of
k. Because
4 St (y, 1i(r2.)) 0
4 D
(x
), then
it would
Tifc Tifc
follow that
e
6"j
(x
nj
) = tf(x
,X\D
nj
CX\H(x
contradicting
y £ g
Because
ßX
"j
ix
being
y e DnkKJ-
) n g
ni
is compact,
in ßX. For each k£
inclusion
nk
N,z
justified
»j
,X\D
nk
ix
nk
"j
"k
ix
nk
: />/,!"
by conditions
,x
))=X\g
(x
CD
(i) and
"k
nk
). Therefore,
the sequence
£ \x
»j
x
nj
) C X\HÍx
(x
"k
£ D
(ii)
n,
)
"k
ix
nk
) whenever
point,
say
z,
)" C St (y, 11(72,)), the last
above,
Therefore,
z£f)\Stiy,Mink)):k£N\
,x
),
) has a cluster
(x
"k
= [y\,
as required.
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plus
the fact
that
490
R. W. HEATH, D. J. LUTZER, AND P. L. ZENOR
6.4.
Remarks,
E. Michael,
(a) Theorem
which
(b) Each
states
that a stratifiable
of the hypotheses
from Examples
a theorem
p-space
of Theorem
in [2], attributed
to
is metrizable.
6.3 is necessary
as can be seen
7.1 and 7.5.
(c) R. Gittings
X is a p-space
zzzA-space
6.3 generalizes
pointed
out to the authors
can be replaced
[6], or even
that the hypothesis
by the hypothesis
a quasi-complete
that
space
of (6.3)
that
X is an M-space
in the sense
[25], a
of G. Creede
[12].
7. Examples.
7.1.
arily
Example.
separable
Gg-diagonal
even
There
is a monotonically
and hereditarily
and which
normal.
That
The space
so that
space
ordered
Example.
M which
irrationals
There
the set
where
U is open in the usual
of real
set of irrational
to Corollary
numbers
numbers.
Example.
cally normal.
There
stratifiable,
it follows
Example.
normal.
Let
compact
and Hausdorff.
onto map.)
of ßN.
ßN
subset
cannot
normal
space
obtained
of
by topolo-
U U V to be open,
and where
ordered
V is any
space;
The other properties
regular
there
Since
space
according
of M listed
closed
Let
of Y. Then
the space
Y is not monotonically
be monotonically
normal.
space
which
space
X which
X can be taken
regular
spaces
to be
is seminormal.
are not mono-
of Example
7.3.
Since
Y = ßX (cf. (2.3)(c)).
image
that any separable
(Argument:
normal
above
fact
image
a closed
is a countcontinuous
closed
from Proposition
one can deduce
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
compact-
Y be separable,
is a continuous
it follows
From this
Let
whose
ßN —> Y is necessarily
Y = ßX
compact
of ßN, the Stone-Cech
numbers.
f: N —> V be any function
ßf:
space
X is not monotonically
regular
is
is not monotoni-
regular
in [16] that
out to the authors
of natural
which
any countable
Hausdorff
neither
pointed
space
is a countable
2.5 that
are compact
normal,
In particular,
Since
line,
numbers,
it was announced
is a continuous
of the usual
dense
S is clearly
[20].
with the usual
Michael
of the form
X be the countable
Jr.,
ification
able
that
from Theorem
F. G. Slaughter,
space
open sets.
monotonically
all sets
normal.
by topologiz-
5.6, since
of [26] and
M is a generalized
of [24].)
There
X is not monotonically
Hausdorff
obtained
[a, b[ ate basic
of the real
is a countable
(In fact,
in the sense
tonically
line
has a
S x S is not
in [23].
an NQ-space
7.4.
which
product
M is the
by taking
In [15] it is proved
is not stratifiable.
normal),
the space
paracompact,
topology
Clearly
perfectly
from Corollary
and whose
5.6, zM is monotonically
are established
7.3.
follows
The space
is heredit-
Sorgenfrey
and from the results
has a Gg-diagonal,
is not normal.
S which
of the form
is a hereditarily
gizing
above
sets
properties
space,
space
Furthermore,
S is the familiar
S has the indicated
7.2.
(whence
is not metrizable.
ing the set of real numbers
a generalized
Lindeltff
normal
that
image
2.7 that
there
is
491
MONOTONICALLY NORMAL SPACES
no nondegenerate
inal number
space
7-5. Example.
There
hereditarily
yet which
does
LOTS.
listed
that
7.6.
Monotone
space'X
normal
which
card-
Let
open interval
seen
X is a compact,
that
normality
is first
and monotonically
the usual
of X follow
There
ordinals
is a space
normal
of X follows
from the results
X such
for each
of the space
uncountable
normal
and
induced
by
X =
topology
homogeneous,
from Theorem
5-3 and
of [21 ] since
it is easily
are isolated
We topologize
points
and basic
tonically
to (5.6).
according
ent order on X, X is actually
X
which
are less
than
or equal
a way that all
neighborhoods
ordered
(It is interesting
of re copies
is not monotonically
X in such
a generalized
of oj, are sets
space,
to note that,
X is mono-
using
a differ-
a LOTS.)
Fix re > 1 and let Z = X". For 1 <i<»
of z. Let
X" (the product
re £ N and yet
co..
X is clearly
normal
that
X are the ordinals
ordinal
of the form ]a, a>A. Since
coordinate
for every
X is separable.
The points
countable
normal
and is not metrizable.
It is easily
of X) is monotonically
normal.
a Gg-diagonal
properties
Example.
to the first
homogeneous
perfectly
0), (1, 1)} have
order.
nonmetrizable
\s monotonically
is a compact
not have
the lexicographic
Z
).
separable,
([0, l] x |0, 1 })\{(0,
the other
that
m (or even for m = 2
countable,
verified
Z such
and z £ Z let z. denote the z'th
E = (z e Z\ z. = ojj for some
¿j and for z £ E let
E(z) =
\i\ 1 < z < re and z . = oj.}.
We verify
that
Z admits
a function
C C Z be closed and let p £ z\c.
then there
exist
ordinals
Gíp, C) as described
in (2.2)(b).
Let
If p 4 E then let Gíp, C) = |p|. If p e E
u satisfying
(a) újj > a > max|p.|
i 4 E(p)|
(where
we adopt
the convention
that
max (r/>) = 0);
(b) \z £ Z\ z. = p. if i 4 Eip) and u < z . < a>{ if i £ Eíp)\
is a subset
of
z\c.
(The
sets
uct topology
described
of Z.)
in (b) are simply
Define
basic
neighborhoods
u(p, C) to be the first
ordinal
of p in the prod-
satisfying
both
(a)
and (b) and let
Gíp, C) = \z£Z\
That
z. = p. if ijéEÍp)
Gíp, C) satisfies
it is a routine
conditions
but tedious
Z is monotonically
verification
and a(p, C) <z.<a>l
it i £ EÍp)\.
(i) and (ii) of Lemma 2.2(b)
that condition
(iii)
is immediate
is also
satisfied.
and
Thus
normal.
COQ
It follows
because
from Theorem
X is not stratifiable:
4.2 that the space
the singleton
X
\ojA
is not monotonically
is a closed
is not a Gg.
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subset
normal
of X which
492
R. W. HEATH, D. J. LUTZER, AND P. L. ZENOR
BIBLIOGRAPHY
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DEPARTMENT OF MATHEMATICS, UNIVERSITY OF PITTSBURGH, PITTSBURGH,-PENNSYLVANIA 15213 (Current address of R. D. Heath and D. J. Lutzer)
DEPARTMENT OF MATHEMATICS, AUBURN UNIVERSITY, AUBURN, ALABAMA 36830
(Current
address
of P. L. Zenor)
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use