Internat. J. Math. & Math. Scl.
VOL. 12 NO.
(1989) I-8
NOWHERE DENSE SETS AND REAL-VALUED
FUNCTIONS WITH CLOSED GRAPHS
IVAN BAGGS
Department of Mathematics
University of Alberta
Edmonton, Alberta TbG 2GI
Canada
(Received April 19, 1988 and in revised form May I0, 1988)
ABSTRACT.
Closed and nowhere dense subsets which coincide with the points of
discontinuity of real-valued functions with a closed graph on spaces which are not
necessarily perfectly normal are investigated.
regular and normal spaces are characterized.
countable connected Urysohn space
X
Certain
It
G
subsets of completely
is also shown that there exists a
with the property that no closed and nowhere
X coincides with the points of discontinuity of a real-valued
X with a closed graph.
dense subset of
function on
KEYWORDS AND HiRASES.
G
Real-valued functions with closed graphs, nowhere dense sets,
sets and non-perfectly normal spaces.
1980 AMS SUBJECT CLASSIFICATION CODES.
1.
INTRODUCTION.
X and
Let
Y
from X
Y
into
The graph of
{x
f
x[f
Let
denote topological spaces.
the usual topology.
D(f)
54CLO, 54C30, 54C50.
R
All spaces will be assumed to be T
and denote the graph of
is closed if
G(f)
is discontinuous at
f
by
G(f)
denote the real line with
2.
Let
f
{(x,f(x))
is a closed subset of
X x Y.
be a function
X x
Ylx X}.
Let
x}.
It was shown by Thompson [I] that if X is a Balre space and if a function
f:X + R has G(f) closed, then D(f) is a closed and nowhere dense subset of X.
It was proven by Dobbs [2] that, if X is perfectly normal, F C X is closed and of
first category in X if and only if there exists a function f:X + R such that
G(f) is closed and D(f) -F.
Let X be a topologlcal space which is not necessarily perfectly normal. In
this note we will investigate those closed and nowhere dense subsets of X which
coincide with the points of discontinuity of a real-valued function with a closed
graph on X. We will consider this problem when X is either a normal, a
completely regular, a regular or a Urysohn space.
2
I. BAGGS
In what follows we will make use of the following known
PRELIMINARIES.
2.
results
THEOREM 2.1.
D(f)
then
Let
X
If
be a topological space.
X
is closed and of the first category in
THEOREM 2.2.
Let
compact subset of
Y
f:X
Y, then
f
f:X
R
/
has a closed graph,
(Dobbs [2]).
be a function with a closed graph.
-I
(K)
X
is a closed subset of
K
If
is a
(Hamlett and
Herrington [3]).
PROPOSITION 2.3.
h:X + Y
If
g:Y
is a continuous function and if
function with a closed graph, then
Z
is a
has a closed graph (Thompson
h
g
f
+
[ ]).
PROPOSITION 2.4.
G
a compact
F-
that
{x
X
Let
subset of
Xlf(x)
be a completely regular topological space.
X, then there exists a continuous function
O (Gillman and Jerlson [4], page 43).
F
If
f:X
is
R
such
and V c will be used to denote
X, throughout
the closure of V and the complement of V, respectively.
[a,b] will be used to
V
If
is a closed subset of
R.
denote a closed interval in
3.
X IS NORMAL. We will characterize closed and nowhere dense G subsets of a
5
X in terms of the points of discontinuity of a real-valued function
with a closed graph. We will first establish the following lemma.
normal space
Let
LEMMA 3.1.
a function with
X
F
G(f)
F C X
which converges to
X
be closed and nowhere dense in
D(f) -F.
closed such that
x
F,
{f(x)Is
I}
R
be
{xsl= I}
in
and let
If for each net
does not converge in
f:X
/
R, then F
is
G.
Case (i).
PROOF:
Suppose
f
is constant on
F
>
Let
O.
For each
X, let V(f(x)) denote an open interval in R of radius E and centered at
f(x). Select for each x F an open neighbourhood U(x) of x such that
f(U(x) -F)C [Vef(x)]
(This is possible, for otherwise given VE(f(x)), there
would exist for each nelghbourhood U(x) of x an element
YU [U(x) -F] such
/ x
that f(yu)
would
and
This
imply
f(yu / f(x) which is
VE(f(x)).
YU
U.
C
F
Let
F and put
impossible.) Put U
x
).
Clearly
t3FU(x
x
x
c.
vn
V
n
in
{If(x)
R
is a compact subset of
X
n. Put
-i
F, F C [f
for each
constant on
follows that
f(y)
and for some integer
F C G
n
for each
for each
-]u
If(x)
for each
-i
n
-n, f(x)
G
n
(Vn)]C
[f(x) -no,
no
(Vn)]C
[f
+,
f(x) +
hi},
-1
and, by Theorem 2.2, f (Vn) is closed
63 U for n- 1 2,... Since f is
for each positive integer
] t [f(x) + 0
Therefore
n, it follows that
y
F
G
and
no
63
nl
y
f(x) +
63
n-I
G
n
and
F
Let
n.
I
f(x)
n-
is
G
n
C U.
G,
y
no]
U -F.
for x
It
F
Since
since
G
n
is open
n.
F be fixed. Define
Case (ii). Suppose f is not constant on F. Let x I
f(x) otherwise. It follows that
F and g(x)
f(x I) if x
g:X + R by g(x)
G(E) is closed since G(f) is closed and g satisfies the hypothesis of the
NOWHERE DENSE SETS AND REAL-VALUED FUNCTIONS WITH CLOSED GRAPHS
Now by Case (i),
lemma.
F
G
is
and the lemma is established.
[3] gave an example (Example 1.6.1)
Hamlett and Herrington
3
X
to show that if
is the space of all ordinals less than or equal to the first uncountable ordinal,
R, then there does not exist a function
D(
{}.
X.
dense in
G(f)
X
Let
THEOREM 3.2.
F
{f(xa)la
F
PROOF:
{xls
and for every net
C X- F
R
/
such that
which converges to
R.
does not converge in
-
I}
f:X
To show necessity
It follows that there exists a continuous function h:X
R
(urao., [], page 4). n :
by
Xlh(x) O}
,
G.
is
{x
sch ha
O,
g(O)
be closed and nowhere
Sufficiency follows from the proceeding lemma.
F
suppose
I}
e
F C X
if and only if there exists a function
is closed, D(f)
x e F,
with a closed graph such that
be a normal space and let
G6
is
R
f:X
g(x)
that the graph of
Put
otherwise.
Is closed.
f
f- g
Clearly
It follows from Prosltlon 2.3
h.
has the required properties.
f
In Example 4.1 it will be shown that ’normal’ cannot be replaced by ’completely
regular’ in Theorem 3.2. First we will establish two lemmas which will be used in
this example.
LEMMA 3.3.
x,
x
Let
f(y)
has a closed graph and if
X, then there
{(,)l"
and
PROOF:
and
R
f:X
If
nowhere dense in
x}
y=
generality that
D(f)
for all
U
For each open neighbourhood
[D(f)] c
any net in
f(Y(u)
converge
f(x),
to
f, g:X
R
Let
XI
exists a dense subset
Suppose
PROOF:
{xeIs I}
If {xla
C
[D(f)]
I} C X1
x
Ya
Y(U) U
Xs(u)
Such an
and
Y(U)’
R.
f
x
such that
f
U
exists for each
X, and,
f(ys(u) 8)
then either
Clearly
[D(f)]
U
X(u)
(u)
and
+ x
if
then
and
is
ys(u) 8
f(ys(u))
f(x
(u)
is closed, it follows that
lXl glXl,
s.
c
or
does
f(x
(u)
If there
D(f) -D(g).
By Lemma 3.3 there exists a net
D(f).
x
c
such that
be two functions with closed graphs.
X
of
X
in
We may assume without loss of
and that
> If(x) + I for all
Since the graph of
R.
LEMMA 3.4.
R.
which is nowhere dense in
has no convergent subnet in
not
is
such that
R.
x, select
-c.
which converges to
has no convergent subnet in
[D(f>] c
C
If(ys)l
s
of
If(X(U)) _> If(ys(u))
Ys(U) since Ys(U) D(f)
such that
each
{ya}
There exists some net
has no convergent subnet in
D(f)
D(f), where
x
I}
has no convergent subnet in
D(f).
x
{xa]a
exists a net
xs
then clearly x
such that
/
x
f(xs)
and
has no convergent subnet in
R.
{xla
D(g). If
e I} C X
[X1 U V(f>] D,
then D U D(f) is nowhere dense in X. It can be shown as in Lemma 3.3 that there
exists a net
has no
J} C [D U D(f)]cc X 1 such that x 8 x and
for all 8 e J, it follows that
convergent subnet in R. Since f(x
g(x
x
D(g).
{x818
f(xs)
8)
4.
X IS COMPLETELY REGULAR.
valued functions on
respectively.
Let
X
(X)
Let
C(X)
8)
and
G(X)
denote the families of all real
which are continuous and which have a closed graph,
denote the family of all subsets of
X.
If
A C X, let
I. BAGGS
4
AI
A.
denote the cardinality of
The next example shows that Theorem 3.2 does not
hold for completely regular spaces.
EXAMPLE 4.1:
Let
(X,T)
with the tangent disc topology
R21y-- 0.
I(x,y)
L
restricted to
P
disc in
P
Put
denote the upper half of the euclidean plane (y > O)
2
Let
Namely, put p
I( x y) R Y >
X
P t3 L. Let T be the topology on X such that
0.
is the usual euclidean topology.
L
tangent to
(Stein and Seeback [6], page i00).
(L)
F
continuum, and each
2
fJ D
L)
M j
there exists
c,
where
c
D(g).
D(f)
and
/C (X)
c, it follows that if
(L)
exists a family
REMARK 4.2:
X
Let
X
L)
G(X)
G
D(f) -F
U
l(L)l
II
l)(L)l
U
U
X
f
R
g
F
and
F
Therefore, there
c.
F
there does
L
be a closed and nowhere
X.
If
F
U
F,
there does not
F
F
,
x
be a compact
X.
To show sufficiency, note that
F
is of first category since
is of first
is compact and hence a Baire space.
is nowhere dense in
I
of
with a closed graph such that
is of first category in
g(x)
be defined by
f
lJ/l <
and for every
f:X + R
is of second category since
h, then
G(X)
gl(X L)
L). Since
f,g
is completely regular.
exists, by Lemma 2.4 a continuous function
g:R
gl(X
and for every
The necessity follows from Theorem 2.1.
category.
L)
L)
be a completely regular space and let
is any open subset of
Therefore
fl(X
and
D(f) -F.
such that
X.
h:X
Since
R
F
is compact
-i
such that
h
G,
there
F.
Let
(0)
O. By Proposition 2.3, if
It follows that D(f) -F.
O, and g(O)
has a closed graph
It follows from Example 4.1 that the condition ’let F be
compact’ cannot be omitted in Theorem 4.3. Example 1.6.1 by Hamlett and Herrington
It will be shown in
[3] shows that Theorem 4.3 does not hold if F is not
5.
X IS REGULAR.
G.
Example 5.2 that Theorem 4.3 does not hold in general for regular spaces.
The
topology on the space presented in Example 5.2 will be a refinement of the topology
on a nice example of a regular space which is not completely regular that was
constructed by Thomas
[7].
An outline of Thomas’ example follows.
See [7] for
further details and a geometric interpretation.
EXAMPLE 5.1
[(2n, y)
p(2n-
R21t
(THOMAS):
0,+1,+2,+3,..., put L(2n)
0,+I,+2,+_3,..., and k 2,3,4,..., put
2
R
and put T(2n- i, k)
[(2n- i + t,l
and b be two points at ’infinity’. Put
If
R2 I0 < y < 1/2t If
l,k)
i(2n- I, I -kI-)
(0, i-
].
Let
a
x
is closed
D(f) -F.
There exists a function
if and only if
PROOF:
if
X
Let
X.
fl(X
Since
be a topological space and let
such that
THEOREM 4.3.
L
has that property that for every
II
such that
The following results hold when
subset of
is any open
containing
it follows from Lemma 3.4 that there exists a family
f
exist a function
X
Let
X.
which is relatively discrete as a subspace of
(L)l > IC(X- L)l
elements of
G(X)
D
and
is the cardinality of the
D(f) --M, then
such that
such that
f
not exist a function
dense subset of
G(X)
f
L
x
is completely regular and
are both continuous, it follows from Lemma 3.4 that
IC(X-
If
is an open set in
is closed and nowhere dense in
D(f) C L, D(g) C L
such that
(X,)
I(L)I
(X,).
and nowhere dense in
{x
x, then
at
n
n
t-
1
)
NOWHERE DENSE SETS AND REAL-VALUED FUNCTIONS WITH CLOSED GRAPHS
X
L(2n)}
U
n
X
U
U
n--
[T(2n-
U
l,k)]}
l,k)U p(2n-
5
U
x e T(2n- l,k)
for
n
0,+1,+2,..., and
a
then
p(2nx
nelghbourhood of x
l,k),
contains all but finitely many points of T(2n-l,k). If x
(2n,y)
L(2n), then
Topologize
2,3,...,
k
as follows.
If
If
then
x
is open.
a nelghbourhood of
x
consists of all but finitely many points of
y-coordlnate and with x-coordlnate that differs from
and
>
y-coordinate
c
X
Let
D
(X,)
For the space
X
a
x
and
b
is a real
c
X
<
is regular and if
In the following example a topology
(X,’) is regular but not completely
X such that {x} is
there will exist an x
such that
G
(X,’)
but there will not exist a function
EXAMPLE 5.2:
Let
(X,)
1,2,...},
where
G(X)
f
D(f)
such that
be the space of Example 5.1.
let the nelghbourhood base at
{xill
x
be the space of Example 5.1.
will be constructed on
regular.
A
If
a.
It was shown by Thomas [7] that
R is continuous, then f(a)
f(b).
f:X
If
which consists of all points with y-coordlnate
b.
an open set containing
with the same
which consists of all points with
is an open set containing
number, then a subset of
’
X
is a real number, then a subset of
c
X
by less than I.
2n
x
x
i
If
X
x
and
x
Let x
a
i, be a sequence in X which
be as defined in Example 5.1.
# a for all
a,
and let
is discrete in (X,), then put
converges to a in (X,). If for fixed i, x
i
S
for some n and k, put
If for fixed i, x
p(2n-l,k)
i
i
t} T(2n- l,k).
S
If x
(2n,YO) L(2n) for some i and n, then put
i
i
S
{(X, Yo) Xl2n- I < x < 2n + I}. For each basic open nelghbourhood U of a
{xl}.
{xi}
i
in
(X,)
and each sequence
to be an open set containing
A
a.
converging to
Let
and by the new nelghbourhood base of
’
containing
k.
Also
V
a
contains
(X,)
in
define
Ui-I
’
be the topology on
Clearly
a.
T(2n- l,k)
a
t)
’
p(2n -l,k)
D
X
generated by
and every open set
for countably many
contains all but countably many elements of
t]
L(2n).
n
V
in
and
It follows
(X,T’) is regular. (X,T’) is not completely regular
since it can be shown as in [7], that if f:(X,’) + R is continuous, f(a)
f(b).
There does not exist a function g: (X,T’) + R with G(E) closed such
{a} is
that D(g)
{a}. For let g:(X,z’) + R be any function with a closed graph which
c, a
is continuous on X- {a}. It can be shown as by Thomas [7] that g(x)
from the construction that
G.
{a}. Since a is a cluster
constant, for all but countably many elements of X
it follows that any net x
in
(X,’),
sequence
any
of
not
but
point of (X,’)
a
in X which converges to a in r’ has the property that g(x) / c. Since the
graph of g
is closed it follows that
g(a) -c
Therefore Theorem 4.3 does not hold when
X
and
g
is continuous at
a.
is regular.
In this section it will be shown that there exists a
countable Urysohn space X (Example 6.3) with the property that if F is any
non-empty, closed and nowhere dense subset of X, there does not exist a function
F (Proposition 6.6). The space X will
R such that G(f) is closed and D(f)
f:X
be essentially the countable connected Urysohn space constructed by Roy [8].
6.
X IS A URYSOHN SPACE.
I. BAGGS
6
The following proposition will be useful later:
Let
PROPOSITION 6.1.
U F
G
n--2
F’s
where the
n
G
Let
PROOF:
G--.
x
F
X
0
Since for
R
f:X
G
F
and
1,2,3,...
n
G(f)
with
G.
G
I
F
each
f
f:X
subnet of
n.
n’s,
and does not converge in
-.
D(f)
and
x
x=I
{x Is
For each
(
Fn
X
x
#
X
Let
X, then there
n, for n
Let
X, there
does not contain a
R.
be a countable connected
f:X
exists a function
R
T
f
Therefore
Ix
and for each net
f(x)
COROLLARY 6.2.
f(Fn)
X- G.
for infinitely many
f(x)
x, it is easily seen that either
has no convergence subnet in R. Therefore
I}
subset of
Fn
and
+ x
which converges to
{f(xa)la
by
is closed and nowhere dense in
exists a net
{xl= I in G
{xl I for each
If(x)l I is unbounded
R
/
is continuous on
n
such that
continuous at
X,
D(f) -G.
closed and
Define
X- G,
is constant on
f
Since
If
are mutually disjoint, closed and nowhere dense in
n
then there exists a function
0,1,2,...
X.
be an open subset of a topological space
G(f)
X
or
is closed.
G
If
space.
2
such that
is not
in
G(f)
is an open
is closed and
G.
D(f)
PROOF:
Follows immediately from the preceding proposition.
EXAMPLE 6.3:
.|Cnn’+
Let
be a countable collection of pairwise disjoint
subsets of the rational numbers indexed by the set of positive and negative integers
C
such that
is dense in
n
n
0,+I,+2,...}
y
0,+1,+2,+3,...,
U
{m}.
R
for each
>
O.
<
s- c
C
x
If
Y
x
<
s
Let
p
N (p)
{(x,y) e
c
is odd put N (p)
{(x,y)
Xl
+ }.
s
s
If
<
x
n
<
s
+ }.
X
be the topology generated on
>
O.
{(x,n) Ix
X
as follows.
p
If
m, put
p
and
or
n- 1,
}.
Let
T
/
[8].
(X,)
is a
R
such that
F, then
D(f)
is a nowhere dense
Note that since F is nowhere dense in X, F (3
for each integer n and every interval I C R. Also, for each
I N
C2n_l is closed and nowhere dense in X. In what follows Vc(f(x)) denotes an
C2n
C2n
open neighbourhood of
PROPOSITION 6.4.
f(x)
Let
of radius
(X,)
closed and nowhere dense and let
F, then for every integer
restricted to the subspace
subset of
PROOF:
I
y
It wlll be established in
X.
f:X
#.
subset of
D(f)
C
n, x
by the neighbourhood system defined above for all
Proposition 6.6 that if there exists a function
n,
For each p
X,
X and let
(s,n)
Xly- n, n + 1
No(p) {(x,y) X]y >
countable connected Urysohn space.
Let F be a closed and nowhere dense subset of
F
m.
is essentially the space constructed by P. Roy
(X,T)
Cn,
as being ’lines’ in the plane through
is even or zero, put
n
and
X
together with a point at ’infinity’,
define a neighbourhood system for
g
Let
n.
We may visualize
I
(
C2n_ I
I
(
s
in
R.
be the space of Example 6.3, let
f:X
n
C2n_l
/
R
be a function with
C2n_l)
G(f)
and for every open interval
be
closed.
I C R,
If
f
is discontinuous on at most a nowhere dense
with the relative topology.
Suppose on the contrary that for some
C R, f l(l N
I.
I
F C X
n
and some open interval
is discontinuous on a dense subset of the subspace
NOWHERE DENSE SETS AND REAL-VALUED FUNCTIONS WITH CLOSED GRAPHS
I N
7
such
X, we may select x I 1 N
0, choose
> 0 such that
and f(N(x)) C V(f(x)). Select s
II
N(x) C (I I
C2n_l such that
that
is
at
such
every
discontinuous
s
and
U(s) of s
neighbourhood
C2n_l
in (X,T) has the property that U(s)
for
small
sufficiently
N(x)
1
that
C2n_l.
f
D(f)
Since
is nowhere dense in
Let
x.
is continuous at
C2n
>
e
C2n
flll
C2nC
nelghbourhoods of
s.
1,2,...)
Sklk-
sequence
f
Since
l(l I
C2n_l)
such that
s
k
is discontinuous at
I1
C2n_l
for all
s, there exists a
k,
s
k
/
s, and,
{f(Sk)Ik- 1,2,...) has no limit point in R. Select
such
that f(Sm
V(f(x)). Then there exists a sequence
Sm {Sklk- 1,2,...}
{xnln 1,2,...} C N(x) where Xn + Sm and {f(x n) In 1,2,...} converges to a
since
G(f) is closed,
point
p
VE(f(x)).
graph of
f
nowhere dense subset of
integer
p @
Clearly
is closed.
This contradicts the fact that the
C2n_l)
Therefore
f (I N
I
for every open interval
C2n_l
is discontinuous on at most a
I C R
and every
n.
PROPOSITION 6.5.
Let
(X,T)
(f) -F,
then for each integer
a subinterval
12n_ I
PROOF:
N
12n_l
C I
F C X
be the space of Example 6.3, let
f:X + R
closed and nowhere dense and let
x
f(Sm).
be
G(f) closed. If
and for each open interval I C R there exists
n
such that
f
be a function with
is continuous at
x
for each
C2n_ I.
I
Let
be an open interval in
R.
For each integer
n, there exists,
12n_l C I such that fl(12n_l N C2n_l) is
12n_l C2n_l with the relative topology. Since F is
nowhere dense in I
C2n and I C2n_2 for each integer n, we may assume
without loss of generality that 12n_l has been chosen for each n such that f
is continuous at each point of (12n_l
C2n)t) (12n_l N C2n_2).
Let x
let
and
1,2,...) be a sequence in X which
12n_l C2n_l
IXnln
1 2,...
converges to x. We may assume without loss of generality that {x In
N
or
or
is eventually in either 12n_l
12n_l C2n
12n_l C2n_2. If
C2n_l
then
it
follows
is
from the way
in
eventually
{x n In 1,2,...)
12n_l C2n_l
12n_l was selected that f(x n) f(x). Suppose {XnJn 1,2,...} is eventually in
12n_l N C2n. Let > O. Since f is continuous on 12n_l C2n, for each
y
12n_l C2n there exists a > 0 such that f(N(y)) C V (f(y)). Select some
y e 12n_l
C2n such that for all sufficiently small nelghbourhoods N(x) of x,
N(x)
C2nC N(y). {Xnln- 1,2,...) will eventually be in N(y). Since
f(N(y)) C VE(f(y)), it follows that {f(Xn) In- 1,2,...) is eventually in
by Proposition 6.4, an open interval
continuous on the subspace
/
V (f(y))
f
of
eventually in
each
x e
f(x).
f(xn)+
I2n_l N C2n_2 f(Xn)
12n_ 1
Let
(X,)
closed and nowhere dense and let
F, then F
PROOF:
1
2,...)
converges.
Similarly it follows that if
+
f(x).
Therefore
f
Since the graph
{Xn In
1,2,...}
is
for
x
is continuous at
C2n_ I.
PROPOSITION 6.6.
D(f)
f(Xn) In
and we may assume that
is closed
Suppose
be the space of Example 6.3, let
f:X
/
R
be a function with
G(f)
F C X
be
If
closed.
#.
F
is a non-empty, closed and nowhere dense subset of
that there exists a function
f:X
/
R
with
G(f)
closed such that
D(f)
X
and
F.
If
8
f
I. BAGGS
continuous on
X -F
y
y
X- F, it follows that
is constant on
f
is constant on
We may assume therefore that
X.
>
and let
>
there exists a
f(N(y)) C V(f(y)).
N6(y C (I N C2n_l) U
0
0
such that
f
N6(y)
and a neighbourhood
and hence
X -F.
is not constant on
V(fy)).
f()
X
Since
of
y
f
Let
is continuous at
such that
We may assume without loss of generality that
t) (I
(I N
C2n_2 for some interval I C R
C2n
and some
It now follows from repeated applications of Proposition 6.5 and the
definition of open sets in (X,T) that for each m > 2n, there exists an interval
C I such that f(l
Cm) C (f(y)). For each m > 2n, select Ym (Ira Cm)"
m
Then
and f(ym
V(f(y)), for m 2n + I, 2n + 2,... Since
integer
n.
Im
V
Ym
f()
V (f(y))
this would contradict the fact that the graph of
f
is closed.
The proposition follows.
In contrast to the results of Dobbs [2] and Thompson [I] for perfectly normal
spaces, it now follows that if
X
satisfies only the Urysohn separation axiom it is
possible that there may exist a function
D(f)
F
if and only if
F
f:X
R
is a closed subset of
with a closed graph such that
X
with a non-empty interior.
That this is true for the space given in Example 6.3 follows immediately from
Proposition 6.6 and Corollary 6.2.
REMARK 6.7:
Proposition 6.6 does not hold for all countable connected Urysohn
spaces although it does hold for the countable connected spaces constructed by
Kannan [9] and Martin [I0].
It is possible
to redefine the topology on the space in
Example 6.3 such that the space remains connected and Urysohn and
F
is closed and nowhere dense and there does exist a function
R
G(f)
is closed and
f:X
C
CO
I
such that
C_I
D(f) --F.
REFERENCES
i.
2.
3.
4.
Characterizing Certain Sets with Functions Having a Closed
THOMPSON, T.
Graph, Boil. del U.M.I. (4__) 12(1975), 327-329.
DOBBS, J. On the Set of Points of Discontinuity for Functions with Closed
Graphs, Cas. Pst. Mat. IIO(1985), 60-68.
HAMLETT, T.R. and HERRINGTON, L.L. The Closed Graph and P-Closed Graph
Properties in General Topology, Vol. 3, Contemporary Mathematics, Amer. Math.
Soc., Providence, Rhode Island, 1980.
GILLMAN, L. and JERISON, M. Rings of Continuous Functions, Van Nostrand,
Princeton, 1960.
5.
6.
7.
8.
9.
I0.
KURATOWSKI, C. Topology, Vol. I (transl.), Academic Press, New York, 1966.
STEIN, L.A. and SEEBACK, J.A. Counterexamples in Topplogy, Holt, Rinehart and
Winston, Inc., New York, 1970.
THOMAS, J. A Regular Space, not Completely Regular, Amer. Math. Monthly, 76,
(1969), 181-182.
ROY, P. A Countable Connected Urysohn Space with a Dispersion Point, Duke
Math. J., 33(1966), 331-333.
KANNAN, V. A Countable Connected Urysohn Space Containing a Dispersion Point,
Proc. Amer. Math. Soc., 35(1972), 289-290.
MARTIN, J. A Countable Hausdorff Space with a Dispersion Point, Duke Math.
J., 33___(1966), 165-167.