A Hofmann-Mislove Theorem for Scott open sets

0.14 MB·English

by  Matthias Schröder

A Hofmann-Mislove Theorem for Scott open sets
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Matthias Schro¨der, TU Darmstadt
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January 27, 2015
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a
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6 Abstract
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Weconsidertheintersectionmaponthefamilyofnon-emptyω-Scott-

open sets of the lattice of opens of a topological space. We prove that in
]
O a certain class of topological spaces the intersection map forms a contin-
L uousretraction ontothespace ofcountablycompact subsetsofthespace
equippedwith(thesequentialisationof)theupperVietoristopology. This
.
h class consists of all sequential spaces which are sequentially Hausdorff.
t
a
m
1 Introduction
[

1 The Hofmann-Mislove Theorem states that in a sober space X the intersection
v map F 7→ (F) is a bijection between the non-empty Scott-open filters on
T
2
the lattice of open subsets of X and the compact saturated subsets of X (cf.
5
[3]). The natural question arises in which cases (and how) this result can be
4
6 generalised to all non-empty Scott-open collection of opens.
0 We give a positive answer for sequentially Hausdorff sequential spaces. Re-
. member that a space is sequentially Hausdorff, if every convergent sequence
1
0 has a unique limit. Our version of the Hofmann-Mislove Theorem states that
5 the intersection map defined on the family OO (X) of all non-empty ω-Scott-
+
1 open collections of opens is a continuous retraction onto the space K(X) of all
:
v countably-compactsubsets in suchspacesX. The correspondingtopologiesare
i theω-ScotttopologyonOO (X)andthesequentialisationoftheupperVietoris
X +
topologyonK(X). Forthe proofweonly use Dependent Choice(DC), whereas
r
a the classical Hofmann-Mislove Theorem needs the Axiom of Choice (AC).
Beforehandwe summerise in Section 2 some properties of sequentialspaces,
of the ω-Scott topology on open subsets, of the upper Vietoris topology on
countably compact sets, and of sequential Hausdorffness.
1
2 Sequential spaces, hyperspaces, sequentially
Hausdorff spaces
2.1 Sequential spaces and qcb-spaces
Asequentialspace isatopologicalspaceinwhicheverysequentiallyopensubset
is open (cf. [2]. The category Seq of sequentialspaces and continuous functions
as morphisms is cartesian closed. The following lemma explains how function
spaces YX are formed in Seq (see e.g. [1, 4] for details).
Lemma 2.1 LetX,Y besequentialspaces. ThenthefunctionspaceYX formed
in Seq has the set C(X,Y) of continuous functions from X to Y as underlying
set and its topology is equal to both
(a) the sequentialisation of the compact-open topology on C(X,Y) and
(b) the sequentialisation of the countably-compact-open topology on C(X,Y).
The convergencerelation of YX is continuousconvergence, i.e., (f ) converges
n n
tof in YX iff (f (x )) converges to f (x ) in Y whenever (x ) converges
∞ n n n ∞ ∞ n n
to x in X. Equivalently, (f ) converges to f in YX iff the transpose fˆ:
∞ n n ∞
N∞×X →Y defined by fˆ(n,x) = fn(x) is continuous, where N∞ denotes the
one-point compactification of N.
Remember that the sequentialisation of a topology is the family of all sequen-
tially open sets pertaining to that topology. Binary and countable products in
Seq are formed by taking the sequential coreflection (sequentialisation) of the
usual Tychonoff product. We denote the binary product in Seq by X ×Y.
A qcb-space is a quotient of a countably based space. The category QCB of
qcb-spacesformsa cartesianclosedsubcategoryofSeq;countable productsand
function spaces are inherited from Seq (cf. [1, 4]). The same is true for the full
category QCB0 of qcb-spaces that satisfy the T0-property.
2.2 Topologies on open subsets
For a given sequential space X let O(X) denote the family of open subsets of
X. On O(X) we consider four natural topologies. For us, the most important
topology is the ω-Scott topology τ . A subset H ⊆O(X) is called ω-Scott
ωScott
open, if H is upwards closed in the complete lattice (O(X);⊆) and D∩H 6=∅
foreachcountabledirectedsubsetD with D∈H. Thesecondtopologyisthe
S
Scott topology τ on (O(X);⊆) which is defined like the ω-Scott topology,
Scott
exceptforconsideringalldirectedfamiliesofopens,notonlyallcountableones.
Clearly, the ω-Scott-topology refines the Scott topology.
The third topology on O(X) is the compact-open topology τ . A basis
CO
of τ is given by sets of the form K⊆ := {U ∈ O(X)|K ⊆ U}, where K
CO
runsthroughthecompactsubsets ofX. The name“compact-opentopology”is
motivatedbythefactthatitcoincideswiththeusualcompact-opentopologyon
2
thefunctionspaceSX,underthenaturalidentificationofanopensubsetU ⊆X
withitscontinuouscharacteristicfunctioncf(U). HereSdenotestheSierpin´ski
space which has {⊤,⊥} as its underlying set and in which {⊤}, but not {⊥}
is open. The map cf(U): X → S is defined by cf(U)(x) = ⊤ :⇐⇒ x ∈ U.
Obviously, cf : U 7→cf(U) constitutes a bijection between O(X) and SX.
The fourth topology on O(X) is the countably-compact-open topology τ .
ωCO
Itisdefinedlikethecompact-opentopology,exceptforconsideringallcountably
compactsubsetsK. RememberthatK ⊆X iscalledcountablycompact,ifevery
countable open cover of K contains a finite subcover of K.
Clearly, τ ⊆ τ ⊆ τ and τ ⊆ τ ⊆ τ . For hereditarily
CO Scott ωScott CO ωCO ωScott
Lindelo¨f spaces (and thus for qcb-spaces) we have τ = τ and τ =
Scott ωScott CO
τ .
ωCO
Now we present some properties of the convergence relations induced by
thesetopologies. MostofthestatementsofProposition2.2belongtothefolklore
of sequential space, nevertheless it seems worth collecting their proofs here.
Proposition 2.2 Let X be a sequential space.
(1) The convergence relations induced on the set O(X) by the Scott topol-
ogy, by the ω-Scott topology, by the compact-open topology, and by the
countably-compact-open topology coincide.
(2) A sequence(U ) converges to U with respect tothe compact-open topol-
n n ∞
ogy if, and only if,
Wk := \ Un∩U∞ ∈O(X) for all k ∈N and U∞ = [ Wk.
n≥k k∈N
(3) A sequence (U ) converges to U w.r.t. the compact-open topology if,
n n ∞
and only if, for every sequence (x ) converging in X to some element in
n n
U∞ there is some n0 ∈N such that xn ∈Un for all n≥n0.
(4) The sequentialisation of each of the four topologies mentioned in (1) is
equal to τ .
ωScott
(5) The space O(X) equipped with the ω-Scott topology is homeomorphic to
SX via the map cf.
(6) If X is a qcb-space, then the Scott topology on O(X) is a qcb -topology.
0
Proof.
a) To show the only-if-parts of (2) and (3), let (U ) converge to U w.r.t.
n n ∞
the compact-open topology τ . Then the sequence (cf(U )) of charac-
CO n n
teristic functions converges to cf(U ) with respect to the compact-open
∞
topology on SX. By Lemma 2.1, (cf(Un))n converges continuously to
cf(U∞). Therefore the universal characteristic function u: N∞×X → S
3
mapping (n,x) to cf(U )(x) is sequentially continuous. As {∞,n|n ≥
n
k ∈N} is sequentially compact in N∞ for each k and{⊤} is open, the set
Wk =(cid:8)x∈X(cid:12)∀k ≤n≤∞.u(n,x)∈{⊤}(cid:9)
(cid:12)
is open in X (see Lemma 2.2.9 in [4]). Clearly W = U ∩U .
k Tn≥k n ∞
By the continuity of u, for any x ∈ U there is some k such u(n,x) =
∞ x
⊤ = u(∞,x) for all n ≥ k , implying x ∈ W . So (U ) fulfils the
x kx n n≤∞
displayed formula in (2).
Moreover, if (xn)n converges to x ∈ U∞, then there is some n1 ∈ N
such that x ∈ W for every n ≥ n , implying x ∈ U for all n ≥
n kx 1 n n
max{k ,n }. This shows the only-if-part of (3).
x 1
b) Nowweprovethattherighthandsideof (2)impliesthat(U ) converges
n n
to U w.r.t. the ω-Scott topology τ . Let H be an ω-Scott-open
∞ ωScott
set containing U . Since (W ) is an increasing sequence of opens with
∞ k k
U = W , there is some k such that W ∈ H, implying U ∈ H
∞ Sk∈N k 0 k0 n
for all n≥k , as required.
0
c) For any countably compact set K the set K⊆ is obviously an element
of τ . Thus τ refines τ , which in turn refines τ , because
ωScott ωScott ωCO CO
compact subsets are countably compact. So convergence with respect
to τ implies convergence w.r.t. τ , and convergence w.r.t. τ
ωScott ωCO ωCO
implies convergence w.r.t. τ . Furthermore τ refines τ , which
CO ωScott Scott
inturnrefinesτ ,becauseK⊆ isScott-openforeverycompactsubsetK
CO
of X. So convergence w.r.t. τ implies convergence w.r.t. τ , and
ωScott Scott
convergence w.r.t. τ implies convergence w.r.t. τ .
Scott CO
We conclude that all four topologies in (1) induce the same convergence
relation on O(X). This completes the proofs of (1) and (2).
d) Forasequence(U ) ofopensets,therighthandsideofStatement(3)
n n≤∞
is equivalent to saying that (cf(U )) converges continuously to cf(U ),
n n ∞
because in the Sierpinski space S every sequence converges to ⊥. By
Lemma 2.1 the convergence relation induced by the compact-open topol-
ogy on SX is continuous convergence. This proves the if-part of (3).
e) Let H be a sequentially open in the topological space (O(X),τ ).
ωScott
Then H is upwards closed, as U ⊆ V and U,V ∈ O(X) implies that the
constant sequence (V) converges to U w.r.t. τ . Now let (U ) be
n ωScott n n
an increasing sequence of open sets with U := U ∈ H. Then
∞ Sn∈N n
(U ) converges to U w.r.t. the ω-Scott topology, because for every ω-
n n ∞
Scott open set G containing U∞ there is some m∈N such that Um ∈G,
implying U ∈ G for all n ≥ m, because G is upwards closed. As H is
n
sequentially open, we have U ∈H for almost all n. We conclude that H
n
is ω-Scott-open and the topology τ is sequential.
ωScott
f) As the four topologies induce the same convergence relation as τ
ωScott
by (1) and τ is sequential, the sequentialisation of either of them is
ωScott
equal to τ . This shows (4).
ωScott
4
g) The bijection cf : O(X) → SX and its inverse are both continuous func-
tions w.r.t. the corresponding compact-open topologies on the sets O(X)
and SX. Since τωScott is the sequentialisation of τCO by (4) and SX car-
ries the sequentialisation of the compact-open topology on C(X,S) by
Lemma2.1,cf anditsinversearealsocontinuousw.r.t.thispairofsequen-
tial topologies. Hence cf is a homeomorphism between O(X) equipped
with the ω-Scott topology and SX. This shows (5).
h) If X is a qcb-space, then X is hereditarily Lindelo¨f (see [4, Proposition
3.3.1]). Therefore τ =τ . Since (O(X),τ ) is homeomorphic
Scott ωScott ωScott
to SX by (5), it is a qcb -space. This shows (6).
0
(cid:3)
In view of this proposition, we henceforth denote by O(X) the sequential
spaceconsistingoftheopensofX asitsunderlyingsetandcarryingtheω-Scott
topology τ . Note that the ω-Scott topology forms the underlying set of
ωScott
the sequentialspaceO(O(X)), whichitself carriesthe ω-Scotttopologydefined
on the complete lattice (O(O(X));⊆). Moreover, O(O(X)) is a qcb -space, if
0
X is a qcb-space by Proposition 2.2(6).
2.3 The upper Vietoris topology on compact sets
Now we dicuss a natural topology on the set of compact subsets of X, known
asthe upper Vietoris topology. Inviewofthe resultsofSection3itmakessense
toconsiderthe largerfamilyofcountablycompactsubsetsofX. Wedenotethe
latter by K(X). For a qcb -space X, however, K(X) coincides with the family
0
of compact subsets, because X is hereditarily Lindelo¨f.
The upper Vietoris topology is generated by the subbasic open sets (cid:3)U :=
{K ∈ K(X)|K ⊆ U}, where U varies over the open subsets of X. A related
topology is the Vietoris topology. It has as a subbasis the sets (cid:3)U and ♦U :=
{K ∈K(X)|K∩U 6=∅}, where again U ∈O(X).
We do not know whether any of these topologies is sequential. However,
bothtopologiesarecountablybasedandthussequential,providedthatX hasa
countablebasis. By K (X)we denote the space ofcountablycompactsubsets
uV
ofX equippedwiththesequentialisationoftheupper Vietoristopology. IfX is
T , then K (X)isa T -space;ifX isa qcb-space,then K (X)is aqcb-space
1 uV 0 uV
as well (cf. Section 4.4.3 in [4]).
The countably-compact-open topology on the set of continuous functions
fromX toY isdefinedbythesubbasisofopensetsC(K,U):={f ∈YX|f[K]⊆
U}, where K is countably compact and U ∈O(Y). By Lemma 2.1, C(K,U) is
open in the exponential YX formed in Seq. We show that this construction is
continuous.
Proposition 2.3 For two sequential spaces X,Y, the map C is a sequentially
continuous function from K (X)×O(Y) to O(YX).
uV
5
Proof. Let (K ) converge to K in K (X), let (V ) converge to V in
n n ∞ uV n n ∞
O(Y), let (f ) converge to f in YX and let f ∈ C(K ,V ). Since K ⊆
n n ∞ ∞ ∞ ∞ ∞
U := f−1[V ] ∈ O(X), there is some n such that K ⊆ U for all n ≥ n .
∞ ∞ ∞ 1 n ∞ 1
From the fact that (K ) converges to K one can easily deduce that the set
n n ∞
L := K ∪ K is countably compact in X. Hence M := f [L] ⊆ V is
∞ Si≥n1 i ∞ ∞
countably compact in Y. By Proposition 2.2(2) the set W := V ∩ V
m ∞ Tn≥m n
is open in Y for all m ∈ N and V∞ = Sm∈NWm. Therefore there is some
n2 ∈ N with M ⊆ W0 ∪... ∪ Wn2, hence M ⊆ Wn2 and f∞ ∈ C(L,Wn2).
Since the countably-compact-open topology induces the convergence relation
on YX by Lemma 2.1, there is some n3 ∈ N such that fn ∈ C(L,Wn2) for all
n ≥ n . For all n ≥ max{n ,n ,n } we have K ⊆ L, W ⊆ V and thus
3 1 2 3 n n2 n
f ∈ C(L,W ) ⊆ C(K ,V ). We conclude that C(K ,V ) converges to
n n2 n n (cid:0) n n (cid:1)n
C(K ,V ) in O(YX). Hence C is sequentially continuous. (cid:3)
∞ ∞
2.4 Sequentially Hausdorff sequential spaces
We discuss sequentially Hausdorff spaces. A topological space is called sequen-
tiallyHausdorff,ifanyconvergentsequencehasauniquelimit. Hausdorffspaces
aresequentiallyHausdorffandsequentiallyHausdorffspacesareT ,whereasthe
1
conversedoesnotholdin either case(cf. [2]). SequentiallyHausdorffsequential
spaces are called L∗-spaces by R. Engelking in [2].
The following lemma summerises essential properties of sequentially Haus-
dorff sequential spaces.
Lemma 2.4 Let X be a sequential space that is sequentially Hausdorff.
(1) The inequality function neq: X ×X → S defined by neq(x,y) = ⊤ :⇐⇒
x6=y is sequentially continuous.
(2) The map x7→X \{x} is a continuous function from X to O(X).
(3) ThemapH 7→X\ (H)isacontinuousfunctionfromO(O(X))toO(X).
T
(4) Let (H ) converge to H in O(O(X)) and let (x ) converge to x
n n ∞ n n ∞
in X. If xn ∈T(Hn) for all n∈N, then x∞ ∈T(H∞).
Proof.
(1) Let(x ) convergetox and(y ) convergetoy inX. Wehavetoshow
n n ∞ n n ∞
that (neq(xn,yn))n converges to neq(x∞,y∞) in S. The only interesting
case is that there exists a strictly increasing function ϕ: N → N such
neq(x ,y ) = ⊥ for all n ∈ N. Then we have x = y for all
ϕ(n) ϕ(n) ϕ(n) ϕ(n)
n ∈ N. Since X is sequentially Hausdorff, this implies x∞ = y∞, hence
neq(x ,y )=⊥, as required.
∞ ∞
(2) By(1)andbycartesianclosednessofSeq,thefunctions: X →SX defined
by s(x)(y) := neq(x,y) is continuous. Clearly, s(x) is the characteristic
function of the open set X \{x}.
6
(3) BythecartesianclosednessofSeqandby(1),thefunctiont: O(O(X))→
O(X) defined by t(H) := {x ∈ X|(X \ {x}) ∈ H} is continuous. If
x ∈ t(H) then clearly x ∈/ (H). Conversely, if x ∈/ (H), then there
T T
is some V ∈ H with x ∈/ V, implying X \{x} ∈ H, as V ⊆ X \{x}.
Therefore t(H)=X\ (H). Note that this shows that (H) is closed.
T T
(4) Let U := X \{x } for all n ≤ ∞. By (1), (U ) converges to U in
n n n n ∞
O(X). Suppose for contradiction that x ∈/ (H ). Then U ∈ H .
∞ T ∞ ∞ ∞
By Proposition 2.2(3) there is some n0 ∈ N such that Un ∈ Hn for all
n≥n , implying x ∈/ (H ), a contradiction.
0 n0 T n0
(cid:3)
We remark that sequential Hausdorffness is necessary for all statements in
Lemma2.4,takingintoaccountthatStatement(2)comprisesthat{x}isclosed
for all x ∈ X and Statement (3) comprises that (H) is closed for all ω-Scott
T
open sets H.
3 A version of the Hofmann-Mislove Theorem
for ω-Scott open sets
We now formulate and prove our version of the Hofmann-Mislove Theorem for
ω-Scott open sets.
Weequipthesetofnon-emptyω-Scottopensetswiththesubspacetopology
inheritedfromO(O(X))anddenotetheresultingspacebyOO (X). Thisspace
+
is sequential, because the singleton {∅} is closed in the space O(O(X)) which
is itself sequential by carrying the ω-Scott topology (cf. Section 2.2). Recall
thatK (X)isthespaceofcountably-compactsubsetsofX equippedwiththe
uV
sequentialisation of the upper Vietoris topology (cf. Section 2.3).
Theorem 3.1 Let X be a sequentially Hausdorff sequential space. Then the
space K (X) is a continuous retract of OO (X). The map H 7→ (H) is a
uV + T
continuous retraction to the continuous section K 7→{U ∈O(X)|K ⊆U}.
Note that the condition of X being sequentially Hausdorff is essential, see Ex-
ample 3.6.
We prove Theorem 3.1 by a series of lemmas. The following embedding
lemma follows in the case of qcb -spaces from [4, Proposition 4.4.9]. By a
0
sequential embedding we mean a continuous injection that reflects convergent
sequences.
Lemma 3.2 For any sequential T -space X the map e: K (X) → O(O(X))
0 uV
defined by e(K):={U ∈O(X)|K ⊆U} is a sequential embedding.
Proof. ByProposition2.2(4), e(K)isω-Scott-openforallK ∈K (X),hence
uV
e(K)∈O(O(X)).
7
To show that e is injective, let K ,K ∈ K(X) with K 6= K . W.l.o.g. we
1 2 1 2
can assume that there is some x ∈ K \K . Then K ⊆ X \Cls{x}, as K is
1 2 2 2
saturated. Hence X \Cls{x}∈e(K )\e(K ). Thus e is injective.
2 1
Toshowthecontinuityofe,let(K ) convergetoK inK (X)andlet(U )
n n ∞ uV n n
converge in O(X) to an open set U ∈ e(K ). By Proposition 2.2(2) and by
∞ ∞
countable compactness of K∞ ⊆ U∞, there is some k ∈ N such that K∞ ⊆
U ∩U ∈ O(X). This implies that (K ) is eventually in (cid:3)( U ∩
Ti≥k i ∞ n n Ti≥k i
U ). This entails K ⊆ U and U ∈ e(K ) for almost all n. Thus (e(K ))
∞ n n n n n n
converges to e(K ) in O(O(X)).
∞
Conversely,let (e(K )) converge to e(K ) in O(O(X)) and let U be an open
n n ∞
setwithK∞ ∈(cid:3)U. ThenU ∈e(K∞),implyingU ∈e(Kn)foralmostalln∈N
by Proposition 2.2(3). This means that (K ) is eventually in the subbasic set
n n
(cid:3)U. We conclude that (K ) converge to K in K (X). (cid:3)
n n ∞ uV
The following lemma is the key observationfor showing countable compact-
ness of (H).
T
Lemma 3.3 Let X be a sequential space. Let U be open and let (x ) be a
n n
sequence of X. Then (x ) has a subsequence that converges to an element
n n
in U if, and only if, (X \Cls{x }) does not converge to U in O(X).
n n
Proof. If-part: If (X \Cls{x }) does not converge to U in O(X), then by
n n
Proposition 2.2(3) there is a convergent sequences (a ) →a in X such that
n n ∞
a∞ ∈U and an ∈/ X \Cls{xn} for infinitely many n∈N. So there is a strictly
increasingfunctionϕ: N→Nwithaϕ(n) ∈Cls{xϕ(n)}foralln∈N. As(aϕ(n))n
convergestoa andx iscontainedinanyneighbourhoodofa ,(x )
∞ ϕ(n) ϕ(n) ϕ(n) n
converges to a as well.
∞
Only-if-part: Let(x ) beasubsequenceof(x ) convergingtosomeelement
ϕ(n) n n n
x ∈ U. By Proposition 2.2(3) neither (X \Cls{x }) nor (X \Cls{x })
∞ ϕ(n) n n n
converges to U in O(X), because x ∈/ X \Cls{x } for all n∈N. (cid:3)
ϕ(n) ϕ(n)
Lemma 3.4 Let X be a sequentially Hausdorff sequential space. Let (H )
n n
converge to H in OO (X) and let (x ) be a sequence such that x ∈ (H )
∞ + n n n T n
for all n∈N. Then (xn)n has a subsequence converging to some element x∞ ∈
(H ).
T ∞
Proof. Choose some open set U contained in H∞. For any n ∈ N the open
set U := X \{x } = X \Cls{x } is not in H , because x ∈ (H ). By
n n n n n T n
Proposition 2.2(3) applied to X′ := O(X), (U ) does not converge to U in
n n
O(X), because (H ) converges to H in O(O(X)). By Lemma 3.3, (x )
n n ∞ n n
has a subsequence (x ) that converges to some element x ∈U. Applying
ϕ(n) n ∞
Lemma 2.4(4) to (H ) , we obtain x ∈ (H ). (cid:3)
ϕ(n) n ∞ T ∞
Lemma 3.4 implies that H 7→ H maps non-empty ω-Scott-open sets to
T
sequentially compact sets (in sequentially Hausdorff sequential spaces). As se-
quentially compact sets are countably compact (cf. [2]), we obtain:
8
Corollary 3.5 Let X be a sequentially Hausdorff sequential space. Then for
every non-empty ω-Scott-open collection H of opens the intersection (H) is
T
both sequentially compact and countably compact.
The next example shows that sequential Hausdorffness is essentialin Corol-
lary 3.5, even in the case of countably based, locally compact T -spaces.
1
Example 3.6 We consider the space X which has N∪{∞ ,∞ } as its under-
1 2
lying set and whose topology is generated by the basis
B :=(cid:8){n},{a,∞1|a≥n},{a,∞2|a≥n}(cid:12)n∈N(cid:9).
(cid:12)
Clearly,thesequenceofnaturalnumbersconvergesbothto∞ andto∞ inX.
1 2
So all sets in the basis B are compact, implying that X is locally compact and
H :=(cid:8)U ∈O(X)(cid:12)N∪{∞1}⊆U(cid:9)∪(cid:8)U ∈O(X)(cid:12)N∪{∞2}⊆U(cid:9)
(cid:12) (cid:12)
is an ω-Scott-open family of opens. But (H) is not countably compact by
T
being equal to the set N. (cid:3)
Now we show that is even continuous in our case.
T
Lemma 3.7 Let X be a sequentially Hausdorff sequential space. Then the map
: OO (X)→K (X), H 7→ (H), is continuous.
T + uV T
Proof. Supposethecontrary. SinceOO (X)issequential, isnotsequentially
+ T
continuous, hence there is a sequence (H ) convergingto H in OO (X) and
n n ∞ +
an open set U such that T(H∞) ∈ (cid:3)U, but T(Hn) ∈/ (cid:3)U for all n ∈ N.
Therefore there exists some xn ∈ T(Hn)\U for all n ∈ N. By Lemma 3.4,
(x ) has a subsequence (x ) converging to some element x ∈ (H ).
n n ϕ(n) n ∞ T ∞
Since x ∈U, (x ) is eventually in U, a contradiction. (cid:3)
∞ ϕ(n) n
Now it is easy to prove Theorem 3.1.
Proof. (Theorem 3.1) By Lemmas 3.2 and 3.7, the maps e: K 7→ {U ∈
O(X)|K ⊆U} and : H 7→ (H) are continuous. Let K ∈K (X). Clearly,
T T uV
K ⊆ (e(K)). Toshowthe reverseinclusion,letx∈ (e(K)). ThenX\{x}∈/
T T
e(K). Since X is sequentially Hausdorff, X \{x} is open, hence K * X \{x}
and x ∈ K. We conclude (e(K)) = K. Therefore K (X) is a continuous
T uV
retract of OO (X) with as a continuous retraction to the section e. (cid:3)
+ T
Theorem 3.1 implies the following two corollaries.
Corollary 3.8 Let X be sequentially Hausdorff and sequential. Then there is
a continuous retraction from the family of all non-empty ω-Scott-open subsets
of O(X) to the family of all non-empty ω-Scott-open filters of O(X).
Asinqcb-spacescompactnessandcountablecompactnessagreeandω-Scott-
open sets are Scott-open in the open set lattice, we get:
Corollary 3.9 Let X be a sequentially Hausdorff qcb-space. Then (H) is
T
compact for every non-empty Scott-open subset H ⊆O(X).
9
Acknowledgement
IthankMatthewdeBrechtforposingmethequestion. Ihadfruitfuldiscussions
about the subject with Victor Selivanov and Matthew de Brecht.
References
[1] M. Escard´o, J. Lawson, A. Simpson. Comparing Cartesian closed Cate-
gories of Core Compactly Generated Spaces.TopologyanditsApplications
143 (2004), 105–145.
[2] R. Engelking. General Topology. Heldermann, Berlin, 1989.
[3] G. Giertz, K.H. Hofmann, K. Keimel, J.D. Lawson, M.W. Mislove, D.S.
Scott. A compendium of Continuous Lattices. Berlin, Springer, 1980.
[4] M.Schr¨oder.Admissible representations for continuouscomputations.PhD
thesis, Fachbereich Informatik, FernUniversit¨at Hagen, 2003.
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