Internat. J. Math. & Math. Scl.
(1985) 201-203
201
Vol. 8 No.
A CHARACTERIZATION OF CLOSED MAPS
USING THE WHYBURN CONSTRUCTION
YVONNE O. STALLINGS
Department of Mathematics
University of Southwestern Louisiana
Lafayette, Louisiana 70504
(Received May 22, 1984)
ABSTRACT. In this paper
X
function f
we modify the Whyburn construction for a continuous
Y. If the range is first countable, we get a characterization of
closed maps- namely, the constructions are the same if and only if the map is closed.
KEY WORDS AND PHRASES.
osed maps, first countable, Whyburn construction.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODE.
1.
Primary 54A10, 54CI0.
INTRODUCTION.
Let f
X
Y be continuous and let X and Y be Hausdorff.
In
[3]
Whyburn
defined the unified space Z to be the disjoint union of X and Y with a set
open in Z if and only if Q
compact Kc (
on X
U Y
open in Y,
N Y,
by defining
f-l(K)
X is open in X, Q
( is compact.
Y is open in Y, and for any
In this paper we modify the topology
to be open if and only if C)
and for any point p
(
Y,
f-l{p)
X is open in X,
Q is compact.
C)
N Y
is
We denote the
modified Whyburn space by
It is obvious that any set open in Z is open in W.
We will show that if f is
closed, the topologies are in fact the same and if Y is first countable, then Z and
W being the same implies that f is closed.
This will yield the following corollary:
COROLLARY. Any continuous function from
a Hausdorff space into the reals
{or any metric space) is closed if and only if the Whyburn space and the modified
Whyburn space are the same.
Z. PRELIMINARIES.
Arguments similar to those of Whyburn’s show that W is a T topological
space containing X as an open subspace and Y as a closed subspace. However,
Y. 0. STALLINGS
202
just as in the Whyburn space,
Whyburn showed that Z
W need not be Hausdorff.
is Hausdorff if X is locally compact.
Asking when W is I-Iausdorff led to the
following definitions and propositions:
DEFINITION 2. 1.
Then A c- X is fiber
X-- Y be continuous.
Let f
f-l(y)
compact if and only if A is closed and for all y e f(A),
A is compact.
X is locally fiber compact if every point has a neighborhood whose closure
Also,
is fiber compact.
PROPOSITION 2.2.
PROOF. Since A is closed in X,
A) fl Y
(W
f-l(p)
then W
If A is fiber compact in X,
Y is open in Y.
A) fl X
(W
A is open.
is open in X; also
Now let p be any point in Y.
Then
{W A) f-l(p) fl A which is compact since A is fiber compact.
PROPOSITION 2.3. If X is locally fiber compact, then W is Hausdorff.
PROOF. The only interesting case is when p is in X and f(p) q. Since
X is locally fiber compact, there exists a U open in X such that p is in U and
U is a neighborhood of q
U is fiber compact. Hence U is open in W and W
by Proposition Z. 2.
We define, as did Whyburn, a retraction r
identity on Y.
W--Y to be f on X and the
The following results parallel those of Whyburn’s for r
Z
Y.
The proof is omitted.
PROPOSITION 2.4.
The map r
%%r
y is continuous, has compact fibers
and is closed (open) if f is.
The next proposition shows that some of the properties mentioned above
actually characterize the modified Whyburn construction.
This proposition is
similar to a theorem about the Whyburn construction proved by Dickrnan
PROPOSITION 2.5.
a Hausdorff
S
Let r
Y be a retraction with compact fibers from
space onto a regular subspace.
compact subsets of X are closed in S,
Let X
S
Y and f
r
IX"
If fiber
then the modified Whyburn space for
Y is homeomorphic to S.
X
f
[I].
Y. If V is open
Let V be the modified Vhyburn space for f X
-I
-I
in S and p is any point in V 0 Y, then r (p) is compact. But r (p)- V
PROOF.
f
-I
(p)
V and therefore V is open in W.
Now let Q be open in W and let x
Q.
If x
X, then Q
X is an open
set in S and is contained in Q. Suppose x e Q fl Y. Then since Y is regular, we
V of x such that x Vc VcQ fl Y. Let f
Q
B.
-1
Then B is fiber compact and so S B is open in S. Let U (S
B) fl r (V).
I(V)
can find a neighborhood
Then U contains x,
3.
is open in
S, and is contained in
MAIN THEOREM.
We now state and prove the major theorem of this paper which allows us to
determine when W and Z are the same.
CHARACTERIZATION OF CLOSED MAPS
THEOREM 3. I.
Let f
X
203
X and Y be Hausdorff, and
Y be continuous,
Then Z and W are equal if and only if f is a closed
let Y be first countable.
mapping.
PROOF. Assume f is closed, Q is open in W and K is any compact subset of Q
N Y.
Let
f-l(K)
A. Then A is closed and f(A) is a closed subset
Q
of K and hence is compact.
Then
A
f[A
f(A) is a continuous, closed sur-
By [2, Theorem 5.3]
jection with compact fibers and therefore is a perfect map.
Hence Q is open in Z.
A is compact.
Let A be a closed set in X.
Now assume that Z and W are equal.
Suppose that y
Since Y is first countable and Hausdorff,
f{A).
is a limit point of
{yn} c f(A) which converges to
such that f(Xn)
Yn"
there exists a sequence of distinct points
{Xn}
we may choose a sequence
{Xn}.
Let B
X. Since for any
W
f-l(yn)
e f(B),
N
-1( K)
U {y}
{yn}
Suppose that f(x)
n
m
_> N,
Yn
Since
{yn}
% y.
> N such that
Xm
e f
Hence f(x)
-1
B is open in Z.
B)" therefore,
(Z
Hence B has a limit point, say x.
converges to y,
(V). Hence
y.
Z
Then we can find disjoint neighborhoods V of z
However, since x is
e U.
impossible.
z
W,
Z
Since B is also infinite it must have a limit
point, contradicting our assumption.
and U of y.
Since
B is fiber compact and thus
is a compact subset of Y
B is compact.
B)
(Z
{Xn}
B
So
Then B is closed in
Now suppose B has no limit points.
B is open in W by Proposition Z.Z.
Now K
f
Yn
in A
y.
Since
there exists an N such that for every
a limit point of
f(Xm) Ym
B,
we have an integer
is in both U and V which is
BoA and A is closed, x e A and there..
fore f(A) is closed.
Notice that Y being first countable is a necessary hypothesis for the prece-
ding theorem.
Let X.
these X.’s.
The following is an example to illustrate this.
[0, 1)
Let Y
for all
X U p
1,2,3
Then let X be the disjoint union of
where p is not in X.
Define Vc Y to be open if
and only if
1) V is
an
open set contained in X or
2) If p e V, then there exists a finite set of indices such that if
V is compact.
and if
t {i 1,
V
in} then
e {i 1,..., in} then
The inclusion map from X to Y is not closed, Y is not first countable at
Xi
X.1
X.1
p and yet W and Z are the same.
REFERENCES
DICKMAN, R.F., JR., Unified Spaces and Singular Sets for Mappings of
Locally Compact Spaces, Fundamenta Mathematicae, 6Z (1968), 103-123.
Z. DUGUND:[I, J., Topology. Boston, Allyn and Bacon (1966).
3. WHYBURN, G.T., A Unified Space for Mappings, Trans. Amer. Math. Soc.
74 (1953), 344-350.
1.