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N E W Z E A L A N D JO U R N A L O F M A T H E M A T IC S
Volume 28 (1999), 285-297
DO M AINS OF PERFECT LOCAL HOM EOM ORPHISM S
M
athew
T im m
(Received May 1997)
Abstract. The structure of domains of perfect local homeomorphisms is inves­
tigated. Finiteness conditions for domains of perfect local homeomorphisms
are presented. A sufficient condition for all components of the domain of a
perfect local homeomorphism to be open is given.
1. Introduction
In this paper all maps f : X —+Y between topological spaces X and Y are
continuous functions. All one point sets are closed and a neighborhood about x is
an open set that contains x. A map / : X —>Y is a local homeomorphism if for each
x € X there is a neighborhood Ux about x and a neighborhood Vx about y = f(x)
such that the restriction (/ | Ux) : Ux —► Vx is a surjective homeomorphism. A
map f : X —> Y is proper, or compact, if and only if, given any compact subset
K c Y, it follows that f~ 1(K) is compact. A map p : X —>Y is perfect if and only
if, for every y € Y, p~l (y) is compact. Note that it is not assumed that perfect
maps are closed, even though this seems to be fairly common in the literature. The
cardinality of a set S is denoted by |5|.
The main result of Jungck and Timm [3], stated below, is a decomposition
theorem for the domain of a proper surjective local homeomorphism. It is a gen­
eralization of Jungck’s result [2, 2.1] that is a decomposition theorem for point
inverses under proper local homeomorphisms.
Theorem 1.1 ([3, 1.4]). Let X and Y be Hausdorff. Assume that Y is connected.
If p : X —>Y is a proper, surjective, local homeomorphism then X = Xo U ... U X n
X j) : X j —►Y
for some n > 0 , with each X j a component of X . Furthermore (p \
is a proper, surjective, local homeomorphism.
The proof of the above theorem, together with the collection of examples we had
at the time, prompted the question (stated as Question 2.3 in Jungck and Timm
[3]) of whether it was necessary to assume that a surjective local homeomorphism
p : X —>Y be proper, in order to assure that every component of X is open in X . In
addition, one is struck by the fact that both the result from Jungck [2, 2.1] and that
from Jungck and Timm [3, 1.4] can be thought of as giving finiteness conditions
on domains of proper local homeomorphisms. Recent attempts to answer Question
2.3 [3], as well as an attempt to gain a deeper understanding of the ideas presented
in [3] have generated the questions below.
1991 AM S Mathematics Subject Classification: Primary 54C10; Secondary 54E40, 54F15.
Key words and phrases: perfect map, proper map, compact map, finiteness conditions.
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MATHEW TIMM
Question 1.2. Given a map p : X —►Y, what conditions on p, X , or Y (other
than the obvious one) will assure that the domain X is a union of finitely many
components? In particular, to what degree can the requirement of [3, 1.4], that p
be proper, be relaxed? Specifically, if p : X —>Y is a perfect local homeomorphism,
what sorts of finiteness conditions can be obtained on Y? If one cannot find reason­
able conditions under which a non-proper local homeomorphism’s domain must be
a union of at most finitely many components, what can be said about the domain
in this situation?
This paper presents a collection of interesting examples and results suggested by
these examples. In Section 2, what is known about relationships between proper,
perfect, and closed are summarized. In Section 3, the finiteness conditions on X
that follow from the assumption that p : X —►Y is a perfect local homeomorphism
are studied. It is in this context that most of the examples are presented. Also in
this context we develop analogs of the results [2, 2.1] and [3, 2.1] and present a
solution to Question 2.3 [3] for perfect local homeomorphisms. Section 4 contains
a brief investigation of what happens if one attempts to assume that the local
homeomorphism p : X —» Y satisfies some sort of “local properness” condition.
While I am most interested in the situation when p is a local homeomorphism, some
of the results are stated more generally. The reader should compare the finiteness
conditions obtained in this paper to those that can be obtained when one assumes
that the spaces or maps have additional structure imposed upon them. In addition
to the works mentioned above, a reasonable summary of these sorts of results can
be found in McAuley and Robinson [4]. There are illustrations that accompany
several of the examples. All the illustrations are at the end of the manuscript.
I
thank Jerry Jungck and Ollie Nanyes for our frequent conversations about the
material in this paper. Jo Heath and Andrew Lelek are thanked for their readings of
an earlier version of the paper. The referee is thanked for the care which (s)he took
reading the manuscript and in particular for pointing out Example 3.15. Thanks
to Mike McAsey and Alberto Delgado for their assistance with the Tex version of
the manuscript. Finally, thanks to Morgan Rose for taking naps.
2. O n Relationships Between Proper, Perfect, and Closed
After an inspection of the proof of [3, 1.4] one is tempted to include the condi­
tion, as occurs frequently in the literature, that a perfect map be closed. But, if
a map is perfect and closed, then it is proper, so this is too strong a requirement.
It immediately places one in situations already studied in Jungck [2] and Jungck
and Timm [3]. However, for the sake of completeness, this section include a brief
treatment of what is known about the relationships between the three properties:
perfect, closed and proper. The proofs of the results in this section are straight­
forward and are all similar. One of the proofs is included to illustrate the general
method. Related results are contained in the papers by Ho [1] and Palais [5].
Lem m a 2.1. Let p : X —> Y be a map. Then p is closed if and only if for each
y E Y and each open subset U of X such that p-1(y) C U, there is an open V C Y
such that y € V and p_ 1(y) C p-1(V) C U.
Lem m a 2.2. Let p : X —*Y be a perfect closed map. Then p is proper.
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DOMAINS OF PERFECT LOCAL HOM EOM ORPHISM S
Proof. Let K c Y be compact. Let U = {Ua : a E 1} be an open cover of p 1(K).
Since p is perfect, for each y E Y, there is a finite collection Uayl, ... ,UayNy of
U that covers p~1{y). Since p is closed we may apply Lemma 2.1 and for each
y e Y obtain an open set Vy in X , such that p~1(y) C p~1{V) C Uk h ^ a yk- But K
is compact, so there are finitely many t/’s, say yi, ■■■, yM such that Vyi,... , VyM
N
"
covers K. So it follows that U ^=1(Uj=iU ay ) is a union of finitely many sets in U
that covers p~x(K). Therefore, p~1(K) is compact.
□
Note that if p : X —>Y is, in particular, a perfect closed local homeomorphism,
then it is a proper local homeomorphism. It then follows, by Jungck [2, 2.7], that if
X and Y are first countable Hausdorff spaces, Y is connected, and p : X —» Y is a
surjective, perfect, closed local homeomorphism, then X is a finite sheeted covering
space of Y with covering projection p.
This section closes with a list of some of the relationships between the three
concepts: perfect, closed, and proper. Recall that a Hausdorff space X is a k-space
if and only if whenever F C X is such that for every compact K C X , K fl F is
closed, it follows that F is closed. Following the fact are examples that elucidate
these results.
Fact 2.3. Let p : X —> Y be a map.
(a) If p is open (in particular, if p is a local homeomorphism) then p is proper if
and only if it is perfect and closed.
(b) If X is first countable and Y is Hausdorff then p is proper if and only if it is
perfect and closed.
(c) If Y is first countable and Hausdorff then p is proper if and only if it is perfect
and closed.
(d) If X is compact and Y Hausdorff then p is closed. So, if p is perfect, then it is
proper.
(e) (Palais [5]) If Y is a fc-space and p is proper, then p is closed. So, if Y is a
fc-space, then p is proper if and only if it is perfect and closed.
Exam ple 2.4. This example illustrates the necessity of the assumption that p be
closed in Facts 2.3 (a) and (c) and the assumption that Y be Hausdorff in Fact 2.3
(b) and (d). Let Yq = {(0,0)}U {(l/n, 0) : n E N}. Let y* denote a fixed point not in
Yo- Let Y = Yo U {y*}. The basic open sets about points z E Yo are those inherited
from the standard topology on M2. The basic open sets about y* are all those of the
form {y*} U {(1/n, 0) : n > no} for no E N. Now, let Xq = Yo where the topology is
the subspace topology inherited from R 2. Let X\ = {(0, l)}U {(l/n, 1) : n E N} with
the topology it inherits from R 2. Let X = X qU X i. Note that X is first countable,
Hausdorff and that Y is first countable but not Hausdorff. Now define p : X —* Y by
p (l, 0) = y*, p(0,0) = (0,0), andp(l/n, m) — (1/n, 0), for m = 0,1. Note that p is a
continuous, local homeomorphism from X to Y . It is perfect by Lemma 3.1, below.
But, it is not closed since F = X\ is closed while p(F) = {y*} U {(1/n,0) : n E N}
is not closed since it does not contain (0,0) which is a limit point of p(F). Further­
more, p is not a proper map since Yq is compact while p ~ 1 (Yo) = X\(0,1) is not.
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MATHEW TIMM
This example also illustrates that the Hausdorff property is not a topological prop­
erty that is preserved by perfect, local homeomorphisms.
Exam ple 2.5. Let R cf denote the real line with the co-finite topology. Let /cf
denote the open interval (0,1) as a subspace of R cf. Let X be the disjoint union
flcfU/cf. Define p : X —* R c{ by p(x) = x. The map p is proper but it is neither open
or closed since I cf is both open and closed in X yet p(Icf) is neither open or closed
in Rc{- Note p is a locally one-to-one map that is not a local homeomorphism and
that neither the domain or range of p is Hausdorff or first countable.
3. Finiteness C onditions on D om ains of Perfect M aps
Jungck’s result [2, 2.1] giving the finiteness condition on the cardinality of the
inverse image of a point in the range says that if p : X —►Y is a proper surjective lo­
cal homeomorphism between first countable Hausdorff spaces and Y is connected,
then there is a fixed n G N such that for all y G Y, |p—1(y) | = n. That is, in
this situation \p~1(y)\ is globally constant and finite on Y. So Theorem 1.1 and
[2 , 2 .1] give, respectively, the global finiteness condition (that the domain consists
of finitely many components) and the point-wise finiteness condition (that point in­
verses consist of finitely many points) for domains of proper local homeomorphisms
between first countable Hausdorff spaces. The hope was to produce global and
point-wise finiteness conditions for the domains of perfect local homeomorphisms.
The first lemma is the analog of Jungck [2, 2.1]. The examples following it show
that this is the best possible result.
Lem m a 3.1. Let p : X —>Y be a locally one-to-one map. Then p is perfect if and
only if for each y G V, |p—1(y) | is finite. So, a local homeomorphism p : X —>Y is
perfect if and only if, |f>-1(2/)| is finite for all y e 7
Proof. Let p : X —> Y be perfect. Suppose there is a y G Y, such that \p~l {y)\
is infinite. For each, x € p~1{y) there is an open set Ux such that x G Ux and the
restriction (p \Ux) : Ux —►p(Ux) is one-to-one. Since p~l (y) is compact, there
are finitely many x, say x\,... , xn such that UXl,... , UXn cover p~1(y). Hence
some UXi contains infinitely many points of p~1{y). Contradiction. The proof of
the converse is immediate since any finite collection of points is compact in any
topological space.
□
D efinition 3.2. Let p : X —* Y be a map. We say |p-1(y)| is locally constant at
y G Y , if and only if there is an open set U about y and an integer n G N such
that for all z G U, \p~l (z)\ = n. We say that \p~1(y)\ is locally constant on Y if
and only if |p- 1 (f/)| is locally constant at every y G Y . We say |p—1(y) | is locally
bounded at y G Y, if and only if there is an open set U about y and an integer
n G N such that for all z e U , |p- 1 (z)| < n. |p- 1 (y)| is locally bounded on Y if and
only if |p- 1 (y)| is locally bounded at each y G V .
So, while Jungck [2 , 2.1] gives that proper local homeomorphisms between first
countable Hausdorff spaces are such that \p~1(y)\ is finite and globally constant,
Lemma 3.1 only implies that for any perfect local homeomorphism |p_ 1(y)| is finite.
Furthermore, Example 3.3, below, shows that there are surjective perfect local
homeomorphisms between connected metric spaces that have points yo G Y at
which \p~1(y)\ is not locally constant. Indeed, Example 3.4 shows, that there are
DOMAINS OF PERFECT LOCAL HOM EOM ORPHISM S
289
surjective perfect local homeomorphisms between metric spaces that have points
yo G Y at which |p-1(y)| is not even locally bounded.
Exam ple 3.3. Let X = X\ U X 2 U X 3 C I 3 in the subspace topology where
X\ is the unit circle X\ = {(:r,0,z) G R 3 : x2 + z2 = 1} and X 2 is the half­
open interval X 2 = {(0,y, 1) G R 3 : 0 < y < |} and X 3 is the closed interval
X 3 = {(0,y, —1) G R 3 : 0 < y < 1}. Let Y = Y\U Y2 C R 3 where Yi is the unit
circle Y\ = X\ and Y2 is the closed interval Y2 = X 3. Let p : X —>Y be the map
defined by (p |X\) : X\ —> Y\ is the standard double cover of the circle to itself,
(p | X 2) : X 2
Y2 is given by p(0,y,l) = (0,y,-1), and (p | X 3) : X 3 -> Y2
is the identity. Then p is a local homeomorphism and is at most 2-to-l. So, by
Lemma 3.1, p is perfect. However, |p-1(y)| is not locally constant at the point
(0, —1) G Y. To the left of (0, —l), |p_ 1(y)| = 2 while starting at (0, —l)
and proceeding to the right |p-1(y)| = 1. See Figure 3.3.
Figure 3.3
Exam ple 3.4. Let
x = { (x ,i)e
U I I N ( i,n ) G M ^ :- —< x < 0 V 0 < a r <
n
=2
;} )•
Let Y = { ( i , 0 ) g I 2 : - 1 < i < 1 } . Consider X and Y as subspaces of R 2 with
the standard topology. Let p : X —* Y be given by the projection p{x,y) = (x, 0).
Note that p is a surjective local homeomorphism, that for each y G Y, \p~1{y)\ is
finite, and so, by Lemma 3.1, p is perfect. However, |p-1(y)| is not even locally
bounded at (0,0) G Y.
The spaces and maps in Examples 3.3 and 3.4 are very nice. Both maps are
perfect local homeomorphisms. In both cases the range is a compact, connected,
locally compact, locally path connected, semi-locally 1-connected metric space.
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The domains of the maps are locally compact, locally path connected, semi-locally
1-connected metric spaces and in Example 3.3 the domain is even connected. Thus,
adding hypotheses on the domain or range of a perfect local homeomorphism is
unlikely to produce theorems as precise as those produced in [2] and [3] for proper
local homeomorphisms. However, a couple of situations to examine are suggested by
these examples. In neither, of 3.3 or 3.4 is the domain compact. Also, in Example
3.4 the domain is not connected. So there is the possibility that if the domain of
p were required to be connected, a global bound on \p~1(y)\ could be obtained.
Furthermore, one wonders what follows if one assumes that \p~1{y)\ is locally or
globally constant on Y or if one assumes that \p~1(y)\is locally or globally bounded
on Y. The next results and examples deal with these questions. Examples 3.5 and
3.6 show the necessity of both the assumption that X be compact and p : X —►Y
be a local homeomorphism in Theorem 3.7.
Exam ple 3.5. There are perfect maps p : X —►Y between compact metric spaces
that are open and closed and yet are such that \p~1 (y)\ is not globally bounded.
Consider the sets X = {(0, l)}U{(l/n, 1) : n 6 N} and Y = {(0,0) U {(1/n, 0) : n €
N} both with the topology they inherit from the standard topology on M2. Both
are compact. Define p : X —> Y by p(0,1) = (0,0), p (l, 1) = (1,1), p (l/k ,l) =
(1/2,0) for 2 < k < 3, p(l/k, 1) = (1/3,0) for 4 < k < 6 , p(l/fc,1) = (1/4,0) for
7 < k < 10,___ The idea in this example is that the next n points in the first
sequence, X , are mapped onto the n-th point (1,1/n) in the second sequence, Y.
By Lemma 3.1, p is perfect. In fact, p is proper, closed, and open. As n —►+oo,
|p-1(0,1/n) |—> +00. However, p is not a local homeomorphism since there is no
neighborhood of (0 , 1) that maps homeomorphically onto a neighborhood of (0 , 0 ).
Exam ple 3.6. This example describes a situation in which X and Y are connected
and p a perfect local homeomorphism such that |p-1(y)| is unbounded on Y .
Observe that in this situation \p~1{y)\ is locally bounded on Y. The domain X is
a union of arcs giving the tree as is shown in Figure 3.6. Note that the arcs that
are the ends of the branches of the tree are half open. The range Y is the union of
circles as is shown in Figure 3.6. The topology on both X and Y is that inherited
from the standard topology on the plane. The labelings of the arcs in X and in Y
show the local homeomorphism p : X —>Y: all the arcs labeled with the integer n
in the domain are mapped homeomorphically onto the single arc with label n in the
range. The arrows on the arcs indicate the direction in which the identifications
occurs. Observe that if you choose a sequence of points y2n+i £ Y such that y2n+\
is in the interior of the (2n + l)-st arc in the range then for n > 0 , \p~1{yn)\= n + 1
so that \p~1{y)\ is unbounded on Y yet each point z € Y has a neighborhood Vz
about z such that |p_ 1(y)| is bounded on Vz. See Figure 3.6.
Theorem 3.7. Let p : X —>Y be a perfect local homeomorphism. If X is compact,
then \p~1{y)\ is globally bounded on Y . Hence, \p~1{y)\ is locally bounded on Y .
DOMAINS OF PERFECT LOCAL HOM EOM ORPHISM S
291
Proof. Suppose that there is no global bound on \p~1(y)\. Then there is a sequence
of points yn € p (X ) such that for each n € N, |p- 1(yn)| > n. Let So = {yn : n € N}.
Denote the closure of So by So- Then K = p~1(S o) is a closed subset of the
compact space X and is therefore compact. For each x 6 K, there is an open
set Ux about x and an open set Vx about p(x) such that p \Ux : Ux —►Vx is a
surjective homeomorphism. But the collection {Ux : x € K} is an open cover of
the compact set K. So there are finitely many, say UXl,... , UXM, that cover K.
Now choose n > M + 1. Then |p-1(yn)| > n > M + 1. Say p~l (yn ) = {^i, • ,zm,
zm+i, • • • ,zn}- Since n > M + 1, some UXjl contains 2 or more of the Zfc’s. But
for all k, p(zk) = yn and p |UXj, : UXj, —>VXj, is a surjective homeomorphism. So
UXj/ contains at most one Zk- Contradiction.
□
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Figure 3.6
One is tempted to try to prove that if p : X —> Y is a perfect local homeo­
morphism and the range Y is compact, then \p~1(y)\ is bounded on Y without
any additional hypotheses on X . However, Example 3.4 indicates that additional
hypotheses are required. In particular, requiring that X be connected seems promis­
ing. However, as Example 3.8 shows, this is not sufficient.
Exam ple 3.8. Take the range of Example 3.6 and have the radii of the nth circle
be ^ . Add the point to which these circles converge. Let X be the same set as in
Example 3.6 and p : X —►Y the same map. Then X is connected, Y is compact
and p is perfect but \p~1{y)\ is not bounded on Y . Note that p is not surjective.
But, the domain can be modified to alleviate this problem.
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MATHEW TIMM
of Y U a half open axe. Extend p to X ' via the identity on the copy of Y and the
“obvious” map on the interval. See Figure 3.8 below.
2
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Figure 3.8
Theorem 3.9.
(a) Let X be Hausdorff and assume p : X —>Y is a perfect local homeomorphism.
U \p~1{y)\ is locally constant on Y, then p is proper.
(b) (Partial converse to 3.9 (a)). If p is proper, Y is connected, and Y is first
countable then \p~1{y)\ is globally (and therefore locally) constant on Y.
Proof. Suppose that |p-1(?/)| is locally constant on Y. We show that this implies
that p is closed. Let yo £ Y and suppose that there is an open set U C X such that
p- 1(yo) C U. Since p is perfect, |p-1(2/o)l is finite. Say p~x(yo) =
- ,xn}.
Since p is a local homeomorphism, there are neighborhoods Uj about Xj and a
neighborhood Vo about yo such that p \Uj : Uj —* Vo is a surjective homeomorphism
and Uj C U. Since X is Hausdorff we may, without loss of generality, assume
that Uj Pi Uk = 0 for j ^ k. Furthermore, since \p~1(y)\ is locally constant on
Y and since [p—1(2/0) I = n, there is a neighborhood V' about yo such that for
every z £ V ', \p~1(z)\ = n. Take V = Vq C\ V' . Consider p-1(V). Clearly,
P~1{yo) C p ~ 1(V). Furthermore, if z £ V, then \p~1(z)\ = n. But by construction,
for each j, p \Uj is onto Vo, so for each j , there is a Uj £ Uj such that p(uj) = z.
Hence, p~ l{z) = p~ x{z) fl (U”= 1C/j). Therefore, p~ 1(V ) C U. Hence, by Lemma
2.2, p is closed. Therefore, by Lemma 2.3, p is proper. The partial converse stated
in the theorem follows immediately from [3].
□
Exam ple 3.10. Observe that even the partial converse to 3.9 (a) stated in 3.9 (b)
is false if the domain is not Hausdorff. Let X = [0,1] x {0} U {(0,2)}. Put the
standard topology on [0,1] x {0}. Use sets of the form {(0,2)}U{(a:, 0) : 0 < x < a}
for 0 < a < 1 to form a basis for X at (0,2). Note that X is not Hausdorff
as there is no separation of (0,2) and (0,0) by disjoint open subsets of X . Let
Y = [0,1] x {0} with the standard topology. Define p : X —>Y to be the projection
p(xi,x 2) = (#i,0). Then p is a perfect local homeomorphism between connected,
compact spaces that fails to have \p~1(y)\ locally constant at (0,0) £ Y. This
failure is due to the fact that the domain of p is not Hausdorff.
DOMAINS OF PERFECT LOCAL HOM EOMORPHISMS
293
In fact, 3.9 (a) itself is false if the domain is not required to be Hausdorff. Let
X ' = X U (0,1] x {3} where the topology on the half-open interval (0,1] x {3} is
the standard topology. Again, let Y be as above and take p : X ' —* Y to be the
projection p(xi,x2) = (x±, 0). Then p is a perfect local homeomorphism onto a
connected, compact metric space with [p_1(?/)| = 2 for all y G F, However, p is not
proper, since p~1(Y) is not compact.
Exam ple 3.11. The hypothesis that Y is connected is essential in the partial con­
verse of Theorem 3.9. Let Y be the disjoint union Y =
where each Bm is
an open interval with the standard topology. Let X = U+=1(U™= 1Amn) be another
disjoint union of the open intervals Amn. Define the perfect local homeomorphism
p : X —* Y to be any map that maps each Amn homeomorphically onto Bm.
Theorem 3.12. Let X be Hausdorff and p : X —> Y be an open map such that
\p~1{y) I is locally constant and finite on Y. Then p is a proper local homeo­
morphism. In fact, if X and p{X) are first countable Hausdorff spaces then,
p : X —* p(X) is a finite sheeted covering projection.
Proof. Let x G X . Let 2 = p(x). Then there is an n € N and a neighborhood V'
about z such that for all y € V1, \p~1(y)\ = n. Let p~1(z) = {xi = x,x 2, ... ,^ n}Since X is Hausdorff there are neighborhoods Uj about Xj such that Uj f]U'k = 0 ,
if j ^ k. Furthermore, since p is open p{Uj) is an open neighborhood about z. Let
V = V' f l p(U[) n ... n p(U'n) and let Uj = p_ 1(V) fl Uj. Note that for each j, Uj
is an open set about Xj such that the restriction p \Uj : Uj —> V is a surjective
one-to-one open map from the open set Uj to the open set V. Hence, p is a local
homeomorphism. It is perfect by Lemma 3.1 and so is proper by Theorem 3.9.
Therefore, by [3] and [2], if p(X ) is first countable, it follows that p : X —■
>p(X) is
a finite sheeted covering projection.
□
The above results provide a detailed analysis of point-wise and local finiteness
conditions on \p~1(y)\ for perfect local homeomorphisms. The examples given
above, especially Examples 3.3 and 3.4, show that one cannot expect to obtain
a global finiteness condition on the domain of a perfect local homeomorphism.
Thus, the final result, Theorem 3.14, of this section, seems to be the best possible
analog of both Theorem 1.1 and [3, 2.1] that one can obtain for perfect local home­
omorphisms. In addition, Theorem 3.14 provides the partial solution to Question
2.3 in Jungck and Timm [3], mentioned in the introduction. Note that Theorem
3.14 improves on the analog for proper maps since it does not require the domain
and range of the map to be Hausdorff or first countable.
Lem m a 3.13. Let X be a topological space. Then every component of X is open
if and only if for every x £ X there is a neighborhood Nx such that Nx intersects
at most finitely many components of X .
Proof. Assume that every component of X is open. Let x G X . Let Cx denote the
component of X . Then Cx is a neighborhood of x that intersects at most finitely
many components of X . Now assume that for each x G X there is a neighborhood
Nx about x that intersects at most finitely many components of X . Let Co be a
component of X . Let x G Cq. Then there is a neighborhood Nx about x such
that Nx intersects at most finitely many components of X . Say Nx intersects only
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components Co,. • • , Cn. Note Nx = (Nx fl Co) U ... U (Nx ft Cn), a disjoint union.
Now x fi Ci U ... U Cn and since components are closed, Ci U ... U Cn is closed.
Therefore NX\(C\ U ... U Cn) is a neighborhood of x contained entirely in CoTherefore Co is open.
□
Theorem 3.14. Let p : X —> Y be a perfect local homeomorphism. If Y is lo­
cally connected or for each y G Y at which Y is not locally connected there is a
neighborhood V(y) about y such that p~1 (V(y)) intersects at most finitely many
components of X , then every component of X is open.
Proof. Y locally connected and p a local homeomorphism implies that X is also
locally connected. So it follows immediately that every component of X is open.
Now, assume that Y is not locally connected but for each y G Y at which Y is
not locally connected there is a neighborhood V(y) about y such that p -1 (V(y))
intersects at most finitely many components of X . Let xo E X . If X is locally
connected at xq then there is clearly a connected neighborhood Uq about x, and
U0 intersects at most finitely many components of X . On the other hand, suppose
that X is not locally connected at Xo. Then xq has a neighborhood Uq such that
for any neighborhood U with xo G U C Uq, U is not connected. Since p is a local
homeomorphism, there is a neighborhood N q about xo such that the restriction
p | N q : N q —>p (N q) is a homeomorphism onto a neighborhood of yo = p{xo). Let
No = N q H U q. Since p is a local homeomorphism it therefore follows that yo is
point of Y at which Y is not locally connected. Hence, by assumption, there is
a neighborhood Vo about yo = p(xo) such that p_ 1(Vo) intersects at most finitely
components of X . So every point in X has a neighborhood that intersects at most
finitely many components of X . Therefore, by Lemma 3.13, every component of X
is open.
□
There is an elementary example (Example 3.17) that shows that the there are
perfect local homeomorphisms between metric spaces for which components of the
domain need not be open even though the range is connected. The next example
is a modification of Example 3.4 suggested by the referee.
Exam ple 3.15. This example show that that the converse of Theorem 3.14 is not
true. Let X q be the set given by
{(x, 0,0) G R 3 : -1 < x < 1} U {(0,0, z) G R 3 : 0 < z < 1}
U {(0,j/, 1) G R 3 : —1 < y < 0} U
| ^ 0 , - i , ^ G R3 : 0 < z < 1
Let X = Xo U ( U ^ ! {(x,n,0) G R 3 : —- < x < 0V 0 < x < £}). Let Y = Xq.
Consider X and Y as subspaces of R 3 with the standard topology. Let p : X —>Y
be given by p(x,y,z) = (x,y,z), if y < 0, and p(x,y,z) = (x,0,z) if y > 0 . As
with Example 3.4, p is a perfect surjective local homeomorphism, but \p~1(y)\ is
not locally bounded at (0,0,0) G Y, which is not a point of local connectivity of Y.
DOMAINS OF PERFECT LOCAL HOM EOM ORPHISM S
295
An interesting problem is the following:
Problem 3.16. Are there any (somewhat natural) hypotheses that will yield both
necessary and sufficient conditions for every component of the domain of a perfect
local homeomorphism to a connected space to be open?
Exam ple 3.17. For k > 0, let
Xk = { ( * ’ 2* ) : 0 “ X ~ *} U
: 2 * _ 2*+2 < y < 2i* + 2^+2 }*
Let X = {(0,0)} U (U^f^Xfe), with the topology it inherits as a subspace of R 2.
For n > 1, let Yn = {(a;,
: 0 < x < l} . Let Y = {(0,0)} U ( U ^ y „ ) U {(1,2/) :
0 < y < f} with the topology it inherits as a subspace of R 2. Define p : X —» Y
by (1) p(0,0) = (0,0) and (2) if (x,y) e X k then p{x,y) = (:x ,y - £ ) . Then Y is
connected and p is a surjective local homeomorphism that is at most 2-to-l. So p
is perfect. {(0,0)} is a component of X that is not open. Note that neither X or
Y is locally connected at (0,0) and that for every neighborhood V about (0,0) in
Y, p~l (V) intersects infinitely many components of X . See Figure 3.17.
----- 1
P
V
Figure 3.17
4. Locally Proper M aps
Theorems 3.9 and 3.12 are results that show that a perfect local homeomorphism,
or more generally a perfect open map, that have additional “local compactness”
conditions placed on them end up being proper. The last two observations of this
paper deal with two obvious ways to impose “local compactness” conditions on the
map. It was hoped that some such condition on the map p would give that every
component of the domain of p is open and yet not imply that p is proper. The
results of the first attempt to impose such a condition is Fact 4.1. This attempt
also just gives another characterization of proper maps, at least when one assumes
the range is Hausdorff. (Without the assumption that the range is Hausdorff, I
am not sure what happens.) Example 4.2 deals with the standard definition of
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local compactness appearing in the literature and its relationship to Question 2.3
of Jungck and Timm [3].
Fact 4.1. If Y is Hausdorff and p : X —>Y is a map such that for each y E Y there
is a compact set K y, with y in the interior of K y, such that p~1(Ky) is compact,
then p is proper.
Proof. Let K be a compact subset of Y. Then K can be covered by finitely many
of the sets K y i, . . . , K Vn that are mentioned in the statement of Fact 4.1. Since
Y is Hausdorff, for each j = 1,. . . ,n, K CI K yj is a closed and, therefore compact,
subset of Kyj so that p~l (Kyj fl K) is compact. Thus
n
p~\K)
= p -1
M J (K V> n K )
n K))
,k = l
is a finite union of compact sets and is therefore compact.
□
Exam ple 4.2. Recall that a map p : X —► Y is said to be locally compact, or
locally proper, if and only if for every x E X , there is a neighborhood U x about x
such that the restriction p | (cl(Ux)) : cl(Ux) —> p(c\(Ux)) is a proper map. Let Y
be the subspace of R 2 given by
Y =
U {(x, 0 ) : i G [0,1]} U {(0, y) € K 2 : y E [0,1]}.
Let X be the disjoint union of a copy of Y together with the subspace
Z =
| U{(x,0) : x E (0,1)}
Let p : X —> Y be defined by p(x, y) = (x , y) for all (x , y) E X . Then p is a perfect
local homeomorphism from the metric space X onto the connected metric space Y
that is a locally compact map, yet the component{(x, 0) : x E (0,1)} C Z C X
is a component of X that is not open in X . So, even locally proper, perfect local
homeomorphisms from a metric space onto a compact, connected metric space,
need not be such that ever component of the domain is open. The reader should
compare this example to Theorem 3.14 and observe that the failure of the theorem
in this example is due to local connectivity problems in Z.
References
1. C.W. Ho, A Note on proper maps, Proc. Amer.Math. Soc. 51 No. 1 (1975),
237-241.
2. G.F. Jungck, Local Homeomorphisms, Dissertationes Mathematicae, CCIX,
Polska Akademia Nauk, Instytut Matematyczny, Warszawa, 1983.
3. G.F. Jungck and M. Timm, Decompositions of domains of proper local homeo­
morphisms, Topology Proceedings, 17 (1992), 145-150.
DOMAINS OF PERFECT LOCAL HOM EOM ORPHISM S
297
4. L.F. McAuley and E.E. Robinson, Discrete open and closed mappings on gen­
eralized continua and Newman’s property, Can. J. Math. 36 No. 6 (1984),
1081-1112. Correction: Can. J. Math. 37 No. 4 (1985), 767-768.
5. R. Palais, When proper maps are closed, Proc. Amer. Math. Soc. 24 (1970),
835-836.
Mathew Timm
Department of Mathematics
Bradley University
Peoria IL, 61625
USA
[email protected]