Download The greatest splitting topology and semiregularity

The greatest splitting topology and
semiregularity
D. N. Georgiou and S. D. Iliadis
University of Patras,
Department of Mathematics,
265 04 Patras, Greece.
E-mail: [email protected]
E-mail:[email protected]
1
It is well known that (see, for example, [1], [2], and [3])
the intersection of all admissible topologies on the set
C (Y; Z ) of all continuous maps of an arbitrary space Y
into an arbitrary space Z , is always the greatest splitting
topology. However, this intersection maybe not admissible.
[1] H. Render, Nonstandard topology on function
spaces with applications to hyperspaces, Transactions
of the American Mathematical Society, Vol. 336, No. 1
(1993), 101-119.
[2] M. Escardo, J. Lawson, and A. Simpson, Com-
paring cartesian closed categories of (core) compactly
generated spaces, Topology and its Applications 143(2004),
105-145.
[3] D.N. Georgiou, S.D. Iliadis, and F. Mynard, Function Space Topologies. Open Problems in Topology 2,
Elsevier, Edited By Elliott Pearl.
2
In the case, where Y is a locally compact Hausdor
space the compact-open topology on the set C (Y; Z )
is splitting and admissible (see [4], [5], and [6]), which
means that the intersection of all admissible topologies
on C (Y; Z ) is admissible.
[4] R.H. Fox, On topologies for function spaces, Bull.
Amer. Math. Soc. 51(1945), 429-432.
[5] R. Arens, A topology for spaces of transformations, Ann. of Math. 47(1946), 480-495.
[6] R. Arens and J. Dugundji, Topologies for function
spaces, Pacic J. Math. 1(1951), 5-31.
3
In [6] an example of a non-locally compact Hausdor
space Y is given having the same property for the case,
where Z = [0; 1], that is on the set C (Y; [0; 1]) the compactopen topology is splitting and admissible. This space Y
is the set [0; 1] with a topology , whose the semi-regular
reduction coincides with the usual topology on [0; 1].
Also, in [6] (Theorem 5.3) another example of a nonlocally compact space Y is given such that the compactopen topology on the set C (Y; [0; 1]) is distinct from the
greatest splitting topology.
4
In this paper, rst we prove that if two spaces (Y; 0)
and (Y; 1) have the same semi-regular reduction, then
for any regular space Z the sets
C ((Y; 0); Z ) and C ((Y; 1); Z )
coincide and a topology t on C ((Y; 0); Z ) is splitting or
admissible if and only if t is respectively splitting or admissible on C ((Y; 1); Z ). Using this fact, we construct
non-locally compact Hausdor spaces Y such that the intersection of all admissible topologies on the set C (Y; Z ),
where Z is an arbitrary regular space, is admissible.
Furthemore, for a Hausdor splitting topology t on
C (Y; Z ) we nd suÆcient conditions in order that t to
be distinct from the greatest splitting topology. Using
this result, we construct some concrete non-locally compact spaces Y such that the compact-open topology on
C (Y; Z ), where Z is a Hausdor space, is distinct from
the greatest splitting topology. Finally, we give some
open problems.
5
1
Preliminaries
Let Y and Z be two spaces. If t is a topology on the
set C (Y; Z ) of all continuous maps of Y into Z , then the
corresponding space is denoted by Ct(Y; Z ).
A topology t on C (Y; Z ) is called splitting if for every
space X , the continuity of a map g : X Y ! Z
implies that of the map gb : X ! Ct(Y; Z ) dened by
relation gb (x)(y ) = g (x; y ) for every x 2 X and y 2 Y .
A topology t on C (Y; Z ) is called admissible if for every
space X , the continuity of a map f : X ! Ct(Y; Z )
implies that of the map ff : X Y ! Z dened by
relation ff(x; y ) = f (x)(y ) for every (x; y ) 2 X Y (see
[5], [4], and [6].)
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Let t be a topology on C (Y; Z ). If a net ff; 2 M g
of C (Y; Z ) converges topologically to an element f of
t
C (Y; Z ), then we write ff; 2 M g !
f . (If M is the
set ! of all non-negative integers, then the net is called a
sequence).
A net ff; 2 M g of the set C (Y; Z ) converges continuously to f 2 C (Y; Z ) (see [7] and [8]) if and only if
for every y 2 Y and for every open neighborhood W of
f (y) in Z there exists an element 0 2 M and an open
neighborhood V of y in Y such that for every 0,
f(V ) W .
[7] O. Frink, Topology in lattices, Trans. Amer. Math.
Soc. 51(1942), 569-582.
[8] C. Kuratowski, Sur la notion de limite topologigue
d' ansembles, Ann. Soc. Pol.Math. 21(1948-49), 219225.
7
The point-open topology (see, for example, [9]) on
C (Y; Z ) is the topology for which the family of all sets
of the form
(y; U ) = ff
2 C (Y; Z ) : f (y) 2 U g;
where y 2 Y and U is an open subset of Z , compose a
subbasis.
[9] J. Dugundji, Topology, Allyn and Bacon, Inc.,
Boston, Mass. 1966.
The compact-open topology (see [4] and [5]) on C (Y; Z )
is the topology for which the family of all sets of the form
(K; U ) = ff
2 C (Y; Z ) : f (K ) U g;
where K is a compact subset of Y and U is an open
subset of Z , compose a subbasis.
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Let O(Y ) be the family of all open sets of the space
Y . The Scott topology on O(Y ) (see, for example, [10]
and [11]) is dened as follows: a subset IH of O(Y ) is
open in the Scott topology if:
() U 2 IH , V 2 O(Y ), and U V imply V 2 IH ,
and
( ) for every collection of open sets of Y , whose union
belongs to IH , there extsts a nite subcollection whose
union also belongs to IH .
[10] G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson,
M. Mislove, and D.S. Scott, A Compendium of Continuous Lattices, Springer, Berlin-Heidelberg-New York
1980.
[11] B.J. Day and G. M. Kelly, On topological quotient maps preserved by pull backs or products, Proc.
of the Cambridge Phil. Soc. 67(1970), 553-558.
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The Isbell topology (see, for example, [12], [13], and
[14]) on C (Y; Z ) is the topology for which the family of
all sets of the form
(IH; U ) = ff
2 C (Y; Z ) : f
1
(U ) 2 IH g;
where IH is a Scott open set of Y and U is an open subset
of Z , compose a subbasis.
[12] P. Lambrinos and B.K. Papadopoulos, The (strong)
Isbell topology and (weakly) continuous lattices, Continuous Lattices and Applications, Lecture Notes in
pure and Appl. Math. No. 101, Marcel Dekker, New
York 1984, 191-211.
[13] R. McCoy and I. Ntantu, Topological properties of spaces of continuous functions, Lecture Notes
in Mathematics, 1315, Springer Verlang.
[14] F. Schwarz and S. Weck, Scott topology, Isbell
topology, and continuous convergence, Lecture Notes
in Pure and Appl. Math. No.101, Marcel Dekker, New
York 1984, 251-271.
10
A subset B of a space X is called bounded (see, for
example, [12]) if every open cover of X contains a nite
subcover of B .
A space X is called corecompact if for every open
neighborhood U of a point x 2 X there exists an open
neighborhood V U of x such that V is bounded in the
space U (see, for example, [10], [12], and [14]).
11
It is well known that:
(1) A topology t on C (Y; Z ) is splitting if and only if
each net of elements of C (Y; Z ) converging continuously
to an element of C (Y; Z ) converges also topologically to
this element (see [6]).
(2) On the set C (Y; Z ) there exists the greatest splitting
topology (see [6]).
(3) The intersection of all admissible topologies coincides
with the greatest splitting topology (see, for example, [1],
[2], and [3]).
(4) The compact-open topology, denoted here by tco, is
always splitting (see [4] and [5]) and, in general, does
not coincide with the greatest splitting topology (see [6]).
For a regular locally compact space Y the topology tco
is always admissible and, therefore, coincides with the
greatest splitting topology (see [4], [5], and [6]).
12
(5) The Isbell topology, denoted here by tIs, is always
splitting (see, for example, [12], [13], and [14]) and, in general, does not coincide with the greatest splitting topology (see [15] and [16]). For a corecompact space Y the
topology tIs is always admissible and, therefore, coincides
with the greatest splitting topology.
(6) If Z is a Hausdor space, then the topologies tco and
tIs are Hausdor (see, [7] and [13]).
[15] M. Escardo, J. Lawson, and A. Simpson, Com-
paring cartesian closed categories of (core) compactly
generated spaces, Topology and its Applications 143(2004),
105-145.
[16] D.N. Georgiou and S.D. Iliadis, On the compact
open and nest splitting topologies. Topology and its
Applications, In Press.
13
A space Y is called Z -harmonic if the compact-open
topology coincides with the greatest splitting topology on
C (Y; Z ). If Y is Z -harmonic for every space Z , then Y
is called harmonic (see [3]).
A space Y is called Z -concordant if the Isbell topology
coincides with the greatest splitting topology on C (Y; Z ).
If Y is Z -concordant for every space Z , then Y is called
concordant (see [3]).
Let (Y; ) be a space. Consider the subset
fInt(Cl(U )) : U
2 g
of . The set b is a base for a topology on Y , denoted
here by sr . The space (Y; sr ) is called semi-regular
reduction of (Y; ). Obviously, sr . A topology is called semi-regular if = sr . Therefore, a topology
is semi-regular if and only if there exists a base for the
topology such that U = Int(Cl(U )) for every element
U of this base. It is easy to verify that: () sr is a
semi-regular topology and ( ) any regular topology is
semi-regular. We say that two spaces (Y; 0) and (Y; 1)
have the same semi-regular reduction if sr0 = sr1 .
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2
The greatest splitting and admissible topologies
In this section we study the relations between the splitting and admissible topologies on two function spaces
C ((Y; 0); Z ) and C ((Y; 1); Z ) in the case, where the
spaces (Y; 0) and (Y; 1) have the same semi-regular reduction.
Proposition 2.1. Let (Y; 0); (Y; 1), and Z be
spaces such that
C ((Y; 0); Z ) = C ((Y; 1); Z )
and for every space X ,
C (X (Y; 0); Z ) = C (X (Y; 1); Z ):
Then, a topology t on C ((Y; 0); Z ) is splitting (respectively, admissible) if and only if t is splitting (respectively, admissible) on C ((Y; 1); Z ).
Proposition 2.2. Let (Y; 0), (Y; 1) be two spaces
with the same semi-regular reduction and Z a regular
space. Then,
C ((Y; 0); Z ) = C ((Y; 1); Z )
and for every space X ,
C (X (Y; 0); Z ) = C (X (Y; 1); Z ):
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(1)
(2)
Let (Y; 0), (Y; 1) be two spaces
with the same semi-regular reduction and Z a regular space. Then, a topology t on C ((Y; 0); Z ) is
splitting (respectively, admissible) if and only if t is
splitting (respectively, admissible) on C ((Y; 1); Z ).
Therefore, the intersection of all admissible topologies
on C ((Y; 0); Z ) is admissible if and only if the intersection of all admissible topologies on C ((Y; 1); Z ) is
admissible.
Proposition 2.4. Let Y be a space whose the
semi-regular reduction is a regular locally compact
space and Z an arbitrary regular space. Then, the
intersection of all admissible topologies on C (Y; Z ) is
admissible and, therefore, the greatest splitting topology is admissible.
Corollary 2.3.
The following proposition is a generalization of Proposition 2.4.
Let Y be a space whose the
semi-regular reduction is corecompact and Z an arbitrary regular space. Then, the intersection of all
admissible topologies on C (Y; Z ) is admissible and,
therefore, the greatest splitting topology is admissible.
Proposition 2.5.
16
Similarly to the above two propositions we can prove
the following propositions.
Let Y be a space whose the
semi-regular reduction Y sr is harmonic and Z an arbitrary regular space. Then, the greatest splitting
topology on C (Y; Z ) is the compact-open topology
on C (Y sr ; Z ).
Proposition 2.7. Let Y be a space whose the
semi-regular reduction Y sr is concordant and Z an
arbitrary regular space. Then, the greatest splitting topology on C (Y; Z ) is the Isbell topology on
C (Y sr ; Z ).
Proposition 2.6.
17
1. Let [0; 1] be the closed interval
of the real line and let M = f1=n : n 2 ! g. Denote
by Y the closed interval [0; 1] with the topology 1 for
which the family 0 [ fY n M g; where 0 is the usual
topology of [0; 1], compose a subbase. By Proposition
2.5, the greatest splitting topology on the set C (Y; [0; 1])
is admissible (see also Theorem 6.21 of [6]).
2. Let (Y; 0) be an arbitrary space and M a subset of
Y with the property that Cl(Y n M ) = Y . On the set Y
we consider the topology 1 for which the family 0 [fY n
M g compose a subbase. The spaces (Y; 0) and (Y; 1)
have the same semi-regular reduction. By Proposition
2.5, for every corecompact space (Y; 0) and an arbitrary
regular space Z , the greatest splitting topology on the
set C ((Y; 1); Z ) is admissible.
Examples 2.8.
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3. Let R be the set of real numbers. On the set R
we consider a topology 0 for which the set f[a; 1) :
a 2 Rg [ f;g compose a base and a topology 1 for
which the set f(1; a) : a 2 Rg [ f;g compose a base.
Note that the space (R; 0) is corecompact but not locally
compact and the space (R; 1) is locally bounded but not
corecompact or locally compact. It is easy to see that
the semi-regular reduction of these spaces is the set R
with the trivial topology. By Proposition 2.5, the greatest
splitting topology on the set C ((R; 1); Z ), where Z is
an arbitrary regular space, is admissible.
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3
On the splitting topologies
Let t be a splitting Hausdor
topology on C (Y; Z ), where Y and Z are arbitrary
spaces. If there exist a sequence ffi : i 2 ! g of elements of C (Y; Z ), an element f1 of C (Y; Z ), and a
t
point y0 of Y such that: (a) ffi : i 2 ! g !
f1 and
(b) there exists an open neighborhood W0 of f1(y0)
in Z such that for every open neighborhood U 0 of y0
in Y there exists i(U 0 ) 2 ! for which fi (U 0) 6 W0
for every i i(U 0 ) (therefore, t is not an admissible
topology), then t is not the greatest splitting topology
on C (Y; Z ).
Corollary 3.2. Let Y be an arbitrary space. If
there exists a Hausdor space Z such that the compactopen topology t = tco on C (Y; Z ) satises the conditions of the Proposition 3.1, then Y is not harmonic.
Corollary 3.3. Let Y be an arbitrary space. If
there exists a Hausdor space Z such that the Isbell
topology t = tIs on C (Y; Z ) satises the conditions of
the Proposition 3.1, then Y is not concordant.
Proposition 3.1.
20
Let Yi, i 2 ! , be a family of Hausdor
spaces such that Yi \ Yj = ; if i 6= j . Suppose that for
every i 2 ! there exists a lter Fi of non-empty open
sets of Yi with the property that Yi n K 2 Fi for every
compact subset K of Yi. On the set
Example 3.4.
[
Y = ( fYi : i 2 !g) [ f1g;
where 1 is a symbol, we consider a topology for which
a subset V of Y is open if and only if:
() V \ Yi is open in Yi for all i 2 ! and
( ) in the case, where 1 2 V , there exists a nite subset
s of ! such that V \ Yi 2 Fi for all i 2 ! n s.
We note that the space Y has the properties:
(1) Yi is simultaneously open and closed subspace of Y .
(2) If K is a compact subset of Y , then there exists a
nite subset s of ! such that K [fYi : i 2 sg [ f1g.
21
Let Z be an arbitrary Hausdor space containing two
distinct points a and b. Consider the sequence ffi; i 2 ! g
of maps of Y into Z for which fi (y ) = b if y 2 Yi and
fi(y) = a if y 2 Y n Yi. Using the above properties (1)
and (2) one can prove that the the maps fi are elements
of C (Y; Z ) and the sequence ffi ; i 2 ! g converges to f1
in the compact-open topology, where f1 is the element
of C (Y; Z ) dened by condition f1 (y ) = a for all y 2 Y ,
which means that the condition (a) of the Theorem 2.1
is satised. We prove that condition (b) of this theorem
is also satised. Indeed, let W0 be an open neighborhood
of a which does not contain the point b. Consider an
arbitrary open neighborhood U 0 of 1 in Y . Then, there
exists a nite subset s of ! such that U 0 \ Yi 6= ; for
every i 2 ! n s. Setting i(U 0 ) = maxfi : i 2 sg we have
that fi (U 0 ) 6 W0 for every i i(U 0 ) proving condition
(b).
Since Z is a Hausdor space the compact-open topology tco on C (Y; Z ) is a splitting Hausdor topology and,
therefore, by Proposition 3.1, tco is not the greatest splitting topology on C (Y; Z ). By Corollary 3.2, the space Y
is not harmonic.
22
The above example can be considered
as a generalization of the space Y considered in the Theorem 5.3 of [6].
Remark 3.5.
4
Some open problems
Let P be a class of spaces having the
same semi-regurar reduction and Z a regular space. By
Proposition 2.2 and Corollary 2.3 the set C (Y; Z ) and the
greatest splitting topology on this set are independent of
the elements Y of P .
(1) Is the compact-open topology on C (Y; Z ) independent of the elements Y of P ?
(2) Is the Isbell topology on C (Y; Z ) independent of the
elements Y of P ?
(3) Suppose that P contains an element which is Z harmonic (respectively, harmonic). Is any element of P
Z -harmonic (respectively, harmonic)?
(4) Suppose that P contains an element which is Z concordant (respectively, concordant). Is any element of
P Z -concordant (respectively, concordant)?
Problems 4.1.
23
It is known that the compact-open
topology does not coincide with the greatest splitting
topology on the set C (N ! ; N ), as well as, on the set
C (R! ; R), where N is the set of natural numbers with
the discrete topology and R is the set of real numbers
with the usual topology.
(1) Suppose that the space Z is the space N or the space
R. Can be nd a sequence ffi : i 2 !g of elements
of C (Z ! ; Z ), an element f1 of C (Z ! ; Z ), and a point
y0 2 Z ! such that the conditions (a) and (b) of the
Proposition 3.1 to be satised?
(2) Let Y and Z be two spaces such that the compactopen (respectively, the Isbell) topology on C (Y; Z ) is not
the greatest splitting topology. Under what (internal)
conditions on Y and Z the conditions of the Proposition
3.1 (concerning the set C (Y; Z )) are satised?
Problems 4.2.
24