Download Monoidal closed structures for topological spaces

C AHIERS DE
TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE
CATÉGORIQUES
M ARIA C RISTINA P EDICCHIO
FABIO ROSSI
Monoidal closed structures for topological spaces :
counter-example to a question of Booth and Tillotson
Cahiers de topologie et géométrie différentielle catégoriques, tome
24, no 4 (1983), p. 371-376
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Article numérisé dans le cadre du programme
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Vol. XXI V - 4 (1983 )
CAHIERS DE TOPOLOGIE
GÉOMÉTRIE DIFFÉRENTIELLE
ET
MONOIDAL CLOSED STRUCTURES FOR TOPOLOGICAL SPACES:
COUNTER-EXAMPLE TO A QUESTION OF BOOTH AND TILLOTSON
by
Maria Cristina PEDICCHIO and Fabio ROSSI
*
INTRODUCTION.
problem of studying monoidal clo sed structures [2] on the category Top of topological spaces and continuous maps
has interested many authors (see, for instance, Wyler [6], Wilker [5] ,
Greve [3], and Booth &#x26; Tillotson [1]). In particular, the method of Booth
&#x26; Tillotson is to topologize, by an arbitrary class d of spaces, both the
set of continuous maps and the set-product of topological spaces (A-open
topology and d.product, respectively). If A satisfies suitable conditions,
then the d-open topology and the 8-product are related by an exponential
law and determine a monoidal closed structure on Top ([I], Theorem
In
recent
years, the
,
2.6, page 42).
In the list of their
for
sequential
spaces,
examples, the authors study the case, important
of A = {N,oo}, where N 00 is the one-point compact-
ification of the discrete space N of natural numbers. This class does
satisfy the hypothesis of Theorem 2.6 of [I.].
not
question they raise is to determine whether this 8-structure is
monoidal closed on Top ([1], Section 6 (i), page 46).
The aim of our note is to answer this question negatively, proving,
by an example, that the considered structure is not monoidal closed.
To obtain this result we will use some properties, due to Isbell
[4], that we recall in Section 1.
One
1.
NOTATION.
X,
Y,
will
Z
hom ( X , Y ) will indicate the
X
i. e.
*
Y. 1 will stand for
to
a
Work
space
consisting
a
of
set
always
topological
of all continuous functions
singleton
two
denote
space,
points
partially supp orted by the Italian
0,1
C N R .
371
spaces and
(maps) from
and 2 for a Sierpinski space,
with 0, {1}, {0, 7 ! as its
open
sets.
Let
first summarize
us
notions and results stated in
some
[1]
and
open
set
[4] .
Lest 8
DEFINITION
in Z, then
be
an
I. 1
[1].
W(f, U)
arbitrary
If A
E
class of
A, f :
will denote the
topological
A - Y is
a
following
spaces.
map and U is
subset of
an
horn ( Y, Z ) :
By requiring the family of all these sets to be an open subbase,
a topology on hom (
Y, Z , called the Q-open topology.
we
intro-
duce
The
M A( - , - )
function space will be denoted by
bifunctor from To pop X Top to Top .
corresponding
is
a
DEFINITION 1.2
the
set
[1].
We define the
X X Y with the final
(f-product X X A Y
topology
with respect
to
M A ( Y, Z ) ;
of X and
all
Y
incoming
as
maps
of the forms :
It is easy
to see
DEFINITION 1.3
that -
[1].
class if for each AE
neighborhoods
in (i .
which
there i s
(t -
is
a
bifunctor from
The
class 3
A
every
are
point
of A has
closed in A and
1.4
Furthermore the
exponential
is a continuous
bijection.
Top
X
Top
of spaces will be said
are
[1]. I f A is a regular
adjunction
PROPOSITION
an
x
law
372
a
to
to
Top.
be
regular
fundamental system of
epimorphic images
class
a
of spaces
of compact spaces, then
[41. A topology on the set Op Y of open sets of a topological space Y is a topological topology iff it makes finite intersection
and arbitrary union continuous operations.
DEFINITION 1.5
PROPOSITION
F -
1.6 (4]. If
then G (2)= Y* is
G,
F
and G
adjoint endofunctors of Top
topological topology on the set Op Y, where
a
are
Y = F(1).
Conversely, if Y is an arbitrary space and Y* is a topological lapology on Op Y, then there is (up to isomorphism) precisely one pair of adjoint endofunctors
F-l
Y and
G(2)
=
Y*.
Z) is hom ( Y’, Z ) with the initial topology with respect
Moreover G (
all the
functions of the form
where
BU(f)
Now,
F(1) =
Top - Top such that
G:
=
f- 1 (U)
we are
able
to
for each f E hom ( Y, Z) .
to
prove the
following proposition,
of compact spaces, then
on Op Y.
a) For any Y c Top , M(i( Y, 2) is a topological topology Y*
with
b) Mél( Y, Z) is the set hom ( Y, Z) with the initial topology
respect to all the functions
PROPOSITION 1. 7.
where BU(f)
=
c) Let Wo
If A
is
regular
class
f -1(U), f hom(Y, Z).
{Ua} be family o f open
E
=
a
on
and let
Y, Z &#x3E;
Hom ( Y, Z) with respect to all the functions
is the set o f all finite intersec tions of Ua
topology
(see b). 1 f W
set o f all arbitrary
denote the initial
Xua’ Ua E W 0
and W2 is the
a
unions
of
sets
of Z
W0
Va, then the initial topologies
coincide.
d) We topologize the
all the
sets
set
Op ( X X (f Y), the subbasic open
of the form
373
sets
being
We denote this
1f
the
topology by
class (i determines
For any X , Y E
a
monoidal closed
structure
Top ,
on
we
have
Top, then,
ne-
cessarily
PROOF.
a
and b follow from 1.4 and 1.6.
c) It is obvious that
Conversely,
is
let U E
W2 ,
then
.
It follows that
commutative, where
identity function of hom (Y,Z).
Since Y* is a
the union operation U is continuous,
then B U. « 1» is also continuous for each U E W2 . It follows, from th e
universal property of initial topology, that «1&#x3E;&#x3E; is continuous.
A similar proof applies to the family W1 .
d) From 1.3 the exponential law is a continuous bijection
A is the canonical
diagonal map and
topological topology,
From b and c,
M (t( X,
Y*)
«1» the
has the initial
topology
with respect
to
all
the maps
where V is
a
subbasic open
set
of Y*. It is easy
374
to see
that
M (1( X, Y*)
and
X X AY&#x3E;
commutative
2. Let
d
homeomorphic.
diagram :
are
be the class whose
only
We obtain the thesis from the
following
element is
compact-
N 00’
the
one-point
ification of the discrete space N of natural numbers.
In this section
we
shall prove that the
d-open topology and
the
8-
determine a monoidal closed structure on Top ; this gives
product
a negative answer to the
question posed by Booth and Tillotson in [1 ,
Section 6, Example ( i ) page 46.
Clearly, for 1.7 d it suffices to find two spaces X and Y such that
do
not
I
Noo, it is easy to see that N, x (i N. = N 00 x N 00 ’ where
x is the topological product. From the definition of the A-open topology,
the open sets of (Noo x Noo) * are generated by the sets of the form
Let X = Y =
where s is any continuous map
Noo- Noo.
For 1.7 d the subbasic
Noo - Noo x Noo’ with comp onents
open sets of Noo x Noo&#x3E; are all the
r, t:
sets
of the form
where
r , t E hom ( Noo, Noo) .
ever, it is not open in
(Noo x Noo)*
ider
a
is
clearly
for every non-empty open
set in
one-point
Th e
( Noo x Noo )*,
contains elements other than
basic open
set
N DO x N 00.
To
see
this,
cons-
m
is 00.
set
Choose m E N such that each
ri (oo)
for 1
375
i
b is either
or
Th en each
has
only
some
n
a
finite number of
in
points
common
E N such that ( m , n ) lies in
contains
N.)Q x Noo / I (m, n )} .
This
no
{m} J x N 00 .
with
Thus the open
si ( Noo) .
completes
the
So there is
proof.
set
I
0
li-structure, for A = {Noo}, is not monoidal closed it fold-product is not associative and 0-1 is not continuous; fur-
Since the
lows that the
ther, M A ( Y, Z) ( cs-open topology
is generally different from
topology)
8, Example (VII),
convergent sequence-open
=
([1],
MCHC ( Y, Z )
Section
page 50 and Section 9 ( i ) page 52).
The authors would like
suggested
to
thank Max
to
Kelly
for the
improvements
the first version of the paper.
REFERENCES.
J. TILLOTSON, Monoidal closed, cartesian closed and convencategories of topological spaces, Pacific J. Math. 88 ( 1980), 35 - 5 3.
1. P. BOOTH &#x26;
ient
KELLY, Closed categories, Proc. Conf.
Algebra (La Jolla, 1965 ), Springer 1966, 421- 562.
2. S. EILENBERG &#x26; G. M.
3. G. GREVE, How many monoidal closed
Math. 34, Basel ( 1980 ), 538-539.
4.
J.
R.
ISBELL, Function
spaces and
structures
categories
there in
Categorical
Top :
Arch.
adjoints, Math. Scand. 36 (1975 ), 317-339.
5. P. WILKER, Adjoint product and hom functors in
Math. 34 (1970), 269- 283.
6. O. WYLER, Convenient
225-242.
are
on
for
general topology, Pacific J.
topology, Gen. Top. and Appl. 3 (1973),
Istituto di Mate mati ca
Universita di Trieste
Piazzale Europa 1
34100 TRIESTE. IT AL IE
376