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C AHIERS DE
TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE
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S. B. N IEFIELD
A note on the locally Hausdorff property
Cahiers de topologie et géométrie différentielle catégoriques, tome
24, no 1 (1983), p. 87-95
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Vol. XXIV-1 (1983)
CAHIERS DE TOPOLOGIE
ET
GÉOMÉTRIE DIFFTRENTIELLE
A NOTE ON THE LOCALLY HAUSDORFF PROPERTY
by
S. B. NIEFIELD
INTRODUCTION.
If X and Y are
the
set
of
objects
from X
morphisms
to
of
a
category
A,
A(X, Y ) denotes
then
Y in A .
object Y of A is cartesian if the functor - X Y : A -&#x3E; A has a right adjoint, i. e. there is a functor ( )Y : A + A together with bijections
8 X, Y : A( X x Y, Z ) -&#x3E; A ( X , ZY)
natural in X and Y. A is said to be cartesian closed if every object of
Let A be
a
category with finite limits. An
A is cartesian.
objects in many non-cartesian closed categories have
been studied, for example, topological spaces by Day and Kelly [2], uniform spaces by Niefield [12], locales by Hyland [6], and toposes by
Johnstone and Joyal [9].
If T is a fixed object of a category A , then A/T is the category
whose objects are morphisms Y + T of A , and morphisms are commutative
triangles in A over T. If A has finite limits, then so does A/ T ; thus,
one can also study cartesian objects over T . This has been done for topological spaces by Booth and Brown [1] and by Niefield [12], and for uniform spaces, affine schemes, and certain locales and toposes by Niefield
[12, 13, 14].
Cartesian
In this paper
consider
we
a
property of
base
T that in-
object
A/ T , provided
a
Y is cartany morphism Y -&#x3E; T is cartesian in
esian in A . We begin with a general result (Theorem 2.1) which is later
interpreted in the categories of topological spaces (Section 3), locales
sures
that
(Section 4) and toposes (Section 5). In view of the fact that
press the
to
be
a
projection
Y -&#x3E;
T, and speak only of objects
desirable property
to
require
of
87
a
over
reasonable base
we
tend
T,
this
object.
to
sup-
seems
2. CARTESIAN DIAGONALS.
Let A be
of A . If Y is
where
an
category with finite limits, and let T be
a
object
denotes the
(Y, P)
T’,
over
back
pulling
along
any
and cartesianness is
graph of p, and rr2
a
morphism
transitive
(via p) provided that
p ). But, again by [12], 1.4,
over T X T (via the diagonal)
is
a
factor its
pullb ack. Thus,
the
always
diagram
then Y is cartesian
Y is cartesian
case
graph of
if T is cartesian
since the
obtain the
over
T (since
Y X T (via the
following
theorem.
Y is cartesian in A, and T has
1f
THEOREM 2.1.
we
this is
projection.
over
objects [l2], 1.4),
relation [12], 1.3; hence,
over
object
as
is the second
preserves cartesian
Y is cartesian
T
fixed
projection p
suppose Y is cartesian in A . Then Y X T is cartesian
Now,
over
we can
a
T via any
Y 4 T
morphism
a
of
cartesian
diagonal,
A.
3. TOPOLOGICAL SPACES.
Top denote
Let
maps, and let T be
a
the category of
fixed
topological
DEFINITION 3.1. A subset A of T is
topological
spaces and continuous
space.
locally
closed if A
=
U n F, where
U is open and F is closed in T.
It is
T iff A
sure
=
an
easy exercise
U n A, for
some
to
show that A is
open subset U of
locally
T , where A
a
closed subset of
denotes the clo-
of A in T.
L E MM A 3. 2. The
following
are
equivalent :
( a ) T h as a cartesian di agon al.
(b) AT = {(t, t) I
t E T} is a locally
88
closed subset
of
T X T , i.
e.,
locally
closed
diagonal.
(c)
locally Hausdorf f space, i. e., every point of
Hausdorff neighborhood.
( d ) T is a union of Hausdorff open subspaces.
T has
a
T is a
PROOF. The
equivalence
and
(
c
)
=&#x3E;
(d) is
[12],
( c ) are equivalent.
that
Suppose
A
Tn U ,
for
is
AT
of (a) and (b) is
locally closed
open subset U of T
some
X
T .
2.7
precisely Corollary
easy exercise. We shall show that
an
a
T admits
T, there is
t E
of
(b) and
subset of T x T . Then
Now, if
a
DT =
an
open
subset V of T such that (t, t) E V x V C LL We claim that V is Haus-
dorff. If
t1, t2 c V and t1 # t2, then (t1, t2) E AT (since V X VnAT C AT)
and hence, t I
and t2 can be separated by disjoint open subsets as desired.
suppose every element of T admits
Conversely,
borhood. Then there is
a
that T
U =
Ua Ua . Let
Clearly, AT c AT Tn U .
=
(s, t) E A T n U.
family {Ua}
that
since
of
Ua
is
Ua
such that
neighsuch
subsets,
Ua Ua x Ua. We claim that AT = Tn U.
(s, t) E AT,
for
hence,
Hausdorff
of Hausdorff open
We shall show that if ( s ,
Suppose
a
Hausdorff,
are
then
we
have
and (s, t) E U. Then
some a
there
t) E A T
and
disjoint
(s, t) E V X W . Now, if ( s , t) E
open subsets V and W
AT,
then
V X W
meets
AT contradicting the disjointness of V and W . Therefore, if (s, t) AT,
the n
(s, t) E AT n U ; and the proof is complete.
locally Hausdorff. An example of
a non-Hausdorff such space is obtained by identifying two copies of the
u nit interval [0. 1] along [0. 1 ) , i. e., th e pushout of the inclusion
[ 0 , 1 ) -&#x3E; [0, 1] along itself.
Clearly,
Next,
it is
not
and the
is
we
any Hausdorff space is
interpret
difficult
to
Theorem 2.1 in
show that
ZY
bijection
given by
89
can
Top. If
Y is
a
cartesian space,
be identified with
T’op ( Y, Z ) ,
is the map
where
Thus, the study of cartesian spaces reduces to the well-known problem
of defining a suitable topology for Top( Y, Z) [3]. But, - X Y has a right
iff it preserves
adjoint
Theorem
[4]).
quotient
Such spaces Y
(by Freyd’s Special Adjoint Functor
characterized by Day and Kelly [2] as
maps
were
those for which the lattice 0 ( Y) of open subsets of Y is
a
continuous
[15]). In particular, if Y is locally compact
(i. e., every point has arbitrarily small compact neighborhoods), then Y is
cartesian in Top. Moreover, if Y is Hausdorff, or more generally sober,
then the converse also holds [5].
lattice (in the
sense
3.3. I f Y
ally Hausdorf f, then
is
THEOREM
PROOF. This
of Scott
locally compact (in
Y is cartesian
easily
over
the above sense) and T is loc-
T via any
follows from Theorem 2.1
by
projection.
Lemma 3.2 and the above
remarks.
In the remainder of this section
locally Hausdorff
investigate
some
properties
of
spaces.
Every locally Hausdorff space
PROPOSITION 3.4.
PROOF.
we
Suppose
that T is
of T X T such that
locally Hausdorff,
is
T1.
and U is
an
open subset
AT = 3.T n U .
Suppose that s , t f T and s # t . We
shall show that s admits an open neighborhood that misses t . If ( s , t ) is
not
in AT then s and t can be separated; and we are done. Thus, we
can assume that ( s , t ) f AT. Now, since A T C U , there is an open neighborhood V of s such that V
Therefore, r
is
X V
C U ;
and t E
V , for if
tf
V , then
T1.
Recall that
a
closed subset of
a
space T is irreducible if it
expressed as a union of two closed proper
sure of a point of T is irreducible. We say that
be
cannot
subsets. Note that the cloT’ is
a
sober space if every
nonempty irreducible closed subset is the closure of a unique point of T’ .
Note that sober « lies between) To and Hausdorff, but is incomparable
90
numbers N with the
example of a non-sober T, space is
« complements of finite subsets » topology.
PROPOSITION 3.5.
Every locally Hausdorff space
with
T1.
A well-known
Suppose
PROOF.
that T is
locally Hausdorff.
irreducible closed subset F contains
the natural
is sober.
We shall show that every
at most one
point.
Let U be
an
open
AF = AF n U. If F contains more than
point, then UBAF is nonempty, for if U c AF, then F is discrete, and
hence not irreducible. Let (s, t) E UBAF. Then there
disjoint open
subset of F X F such that
one
are
V and W of s and t ,
neighborhoods
FBW
are
respectively,
proper subsets of F and closed subsets of
contradicting
irreducibility of
the
F.
Therefore,
But, FBV and
in F.
T’, and
T is
a
sober space.
locally compact locally Hausdorff space.
locally compact iff it is locally closed.
PROPOSITION 3.6. L et T be a
Then a
Y
subspace
of
T is
locally compact subspace of a locally Hausdorff
space T . Then the inclusion Y -&#x3E; T is cartesian by Theorem 3.3; hence,
Y is a locally closed subspace of T by [12] , 2.7.
Conversely, suppose Y is a locally closed subspace of T . Then Y
is cartesian over T (via the inclusion) by [12], 2.7, and T is cartesian
in Top by transitivity. But, Y is a sober space by Proposition 3.5, and
every cartesian sober space is locally compact [5]. This completes the
proof.
PROOF.
Suppose
Note that
Y is
we
a
did
not use
the local compactness of T in the first
part of the above proof, i. e., we showed that any
of a locally Hausdorff space is locally closed.
locally
compact
subspace
4. LOCALES.
Let L oc denote the category of locales and
morphisms
in the «geo-
metric) direction.
One
can
sublocales of
a
consider
locale,
locally
and
so
closed (i.
it makes
91
e.
pullbacks
sense to
of open and
talk about
a
closed)
locale whose
diagonal
locally closed. But,
is
if T
is
a
space, then
the
product
O ( T ) X O ( T ) of the locale of opens of T with itself in Loc is not necessarily isomorphic to O(TXT) ; hence, T may be locally Hausdorff,
locally closed diagonal in Loc . Note that
if T is a locally compact sober space, then D (T) x O ( T ) = O ( T x T)
[7], and it follows that T is locally Hausdorff iff O ( T ) has a locally
while O ( T ) does
closed
er to
diagonal.
[8].
have
not
For
a
further discussion of this matter,
a
DEFINITION 4. 1. A locale L
closed
locally
LEMMA
esian
is
locally strongly Hausdorf f
if L has
a
diagonal.
4.2. A locale L is
locally strongly Hausdorf f i f f
L has
a
cart-
diagonal.
P ROO F. This
follows
directly
sublocale is cartesian iff it is
from Definition 4.1 since the inclusion of
a
locale A is cartesian in L oc iff
compact, i. e. a continuous lattice.
Theorem 2.1 and the above lemma we get:
locally
THEOREM 4.3.
dorff in
if A
is
a
locally closed [13].
Now, Hyland [6] showed that
it is
refer the read-
we
locally compact,
Loc, then A is cartesian
over
Putting
this
together
with
and L is
L via any
locally strongly Hausmorphism A -&#x3E; L in Loc .
5. TOPOSES.
B Top/ Sets denote
2-category of Grothendieck toposes
i. e. bounded toposes over Sets , geometric morphisms, and natural transformations. Now, ISets has finite limits [10 ; hence, one can consider cartesian toposes in the appropriate 2-categorical sense.
If S is any topos, then the notion of an open or a closed subtopos,
hence a locally closed subtopos makes sense [10].
Let
the
DEFINITION 5.1. A Grothendieck
if the
diagonal S -&#x3E; S x Ssets S
If A is
is
a
topos S is locally strongly Hausdorff
locally closed inclusion.
locally strongly Hausdorff locale, then the topos Sh A
of set-valued sheaves on A is a locally strongly Hausdorff topos, since
a
92
Sh A -&#x3E; Sh(A x A) is
locally
a
closed
inclusion, and
locally compact sober space, then Sh T is locally
iff T is a locally Hausdorff space. Of course, if T is
not locally compact, then T may be locally Hausdorff without Sh T being
locally strongly Hausdorff.
particular, if T
strongly Hausdorff
In
is
a
LEMMA 5.2. Let S be a Grothendieck
Hausdorff iff
the
topos. Then S is
diagonal S -&#x3E; S x Sets S
is
locally strongly
cartesian inclusion in
a
B Top/S x Sets S.
PROOF. The inclusion of
[13] ; hence,
In
a
is cartesian iff it is
subtopos
the desired result follows.
[9], Johnstone
is cartesian in
and Joyal showed that
B Top/Sets
remarks
leading up
following theorem.
to
iff it is
Theorem 2.1
a
category and S is
metric
a
are
Grothendieck topos E
Noting
valid for B TOP/ Sets
we
toposes. 1 f E is
locally strongly Hausdorff topos,
E -&#x3E; S is cartesian in B
morphism
a
continuous category.
THEOREM 5.4. Let E and S be Grothendieck
uous
locally closed
that the
obtain the
a
contin-
then any geo-
Top/S.
interesting property of locally compact
locally Hausdorff spaces. Although every locally Hausdorff space Y is cartesian as a space, it is not necessarily the case that Sh Y is cartesian
We conclude with another
as
a
Grothendieck topos. The spaces Y such that Sh Y is cartesian
precisely
metastably locally compact spaces [9].
if Y is stably locally compact, i. e.,
the
is cartesian
Ut « V1 andu 2 « V2
in Y
implies
In
particular,
th n U2 « V1 n V2
are
Sh Y
in Y,
where « denotes the usual «way below » relation.
LEMMA 5.3.
Y is a
If
locally compact Hausdorff space,
then Y is
stably
compact space U « V iff there is
a com-
locally compact.
PROOF. Recall that in
a
locally
pact subset C such that U C C C V
93
[5]. Now,
suppose
and
Then there
are
compact subsets
in
and
C,
C2
such that
and
But, C1 n C2
is compact since Y is
Hausdorff,
Therefore, Vi n V2 « V1 n V2 ; and
and
it follows that Y is
stably locally
compact.
THEOREM 5.4.
Y is
1f
Sh Y is cartesian in
open
cover
a
stable local compactness is
open
cover
then
locally compact locally Hausdorff space, then Y has
consisting of locally compact Hausdorff spaces. Now, meta-
PROOFo If Y is
an
locally compact locally Hausdorff space,
B Top / Sets,
a
consisting
a
local property ( Y satisfies it iff Y has
of such spaces
from the above lemma.
94
[9]); hence,
an
the result follows
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the compact-open
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SCHENECTADY,
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U. S. A.
95