arXiv:1201.1568v1 [math.GN] 7 Jan 2012
The C-compact-open topology on function spaces
Alexander V. Osipov
Ural Federal University, Institute of Mathematics and Mechanics, Ural Branch of the
Russian Academy of Sciences, 16, S.Kovalevskaja street, 620219, Ekaterinburg, Russia
Abstract
This paper studies the C-compact-open topology on the set C[X] of all real-
valued continuous functions on a Tychonov space Xand compares this topol-
ogy with several well-known and lesser known topologies. We investigate the
properties C-compact-open topology on the set C[X] such as submetrizable,
metrizable, separable and second countability.
Keywords: set-open topology, C-compact subset, compact-open topology,
topological group, submetrizable
2000 MSC: 54C40, 54C35, 54D60, 54H11, 46E10
1. Introduction
The set-open topology on a family λof nonempty subsets of the set X
[the λ-open topology] is a generalization of the compact-open topology and
of the topology of pointwise convergence. This topology was first introduced
by Arens and Dugundji [1].
All sets of the form [F, U ] = {f∈C[X] : f[F]⊆U}, where F∈λand
Uis an open subset of real line R, form a subbase of the λ-open topology.
The topology of uniform convergence is given by a base at each point f∈
C[X]. This base consists of all sets {g∈C[X] : sup
x∈X
|g[x]−f[x]|< ε}. The
topology of uniform convergence on elements of a family λ[the λ-topology],
where λis a fixed family of non-empty subsets of the set X, is a natural
generalization of this topology. All sets of the form {g∈C[X] : sup
x∈F
|g[x]−
Email address: [Alexander V. Osipov]
Preprint submitted to Elsevier January 10, 2012
f[x]|< ε}, where F∈λand ε > 0, form a base of the λ-topology at a point
f∈C[X].
Note that a λ-open topology coincides with a λ-topology, when the family
λconsists of all finite [compact, countable compact, pseudocompact, sequen-
tially compact, C-compact] subsets of X. Therefore C[X] with the topology
of pointwise convergence [compact-open, countable compact-open, sequen-
tially compact-open, pseudocompact-open, C-compact-open topology] is a
locally convex topological vector space.
Moreover, if a λ-open topology coincides with a λ-topology, then λcon-
sists of C-compact subsets of space Xand the space Cλ[X] is a topological
algebra under the usual operations of addition and multiplication [and mul-
tiplication by scalars].
2. Main definitions and notation
In this paper, we consider the space C[X] of all real-valued continuous
functions defined on a Tychonov space X. We denote by λa family of
non-empty subsets of the set X. We use the following notation for various
topological spaces with the underlying set C[X]:
Cλ[X] for the λ-open topology,
Cλ,u[X] for the λ-topology.
The elements of the standard subbases of the λ-open topology and λ-
topology will be denoted as follows:
[F, U ] = {f∈C[X] : f[F]⊆U},
hf, F, εi={g∈C[X] : sup
x∈F
|f[x]−g[x]|< ε}, where F∈λ,Uis an
open subset of Rand ε > 0.
If Xand Yare any two topological spaces with the same underlying
set, then we use the notation X=Y,X6Y, and X < Y to indicate,
respectively, that Xand Yhave the same topology, that the topology on Y
is finer than or equal to the topology on X, and that the topology on Yis
strictly finer than the topology on X.
The closure of a set Awill be denoted by A; the symbol ∅stands for
the empty set. As usual, f[A] and f−1[A] are the image and the complete
preimage of the set Aunder the mapping f, respectively. The constant zero
function defined on Xis denoted by 0, more precisely by 0X. We call it the
constant zero function in C[X].
We denote by Rthe real line with the natural topology.
2
We recall that a subset of Xthat is the complete preimage of zero for a
certain function from C[X] is called a zero-set. A subset Oof a space Xis
called functionally open [or a cozero-set] if X\Ois a zero-set. A family λ
of non-empty subsets of a topological space [X, τ ] is called a π-network for
Xif for any nonempty open set U∈τthere exists A∈λsuch that A⊂U.
Let λ={A:A∈λ}. Note that the same set-open topology is obtained if
λis replaced by λ. This is because for each f∈C[X] we have f[A]⊆f[A]
and, hence, f[A] = f[A]. Consequently, Cλ[X] = Cλ[X]. From now on, λ
denotes a family of non-empty closed subsets of the set X.
Throughout this paper, a family λof nonempty subsets of the set Xis a
π-network. This condition is equivalent to the space Cλ[X] being a Hausdorff
space. The set-open topology does not change when λis replaced with the
finite unions of its elements. Therefore we assume that λis closed under
finite unions of its elements.
Recall that a subset Aof a space Xis called C-compact subset Xif, for
any real-valued function fcontinuous on X, the set f[A] is compact in R.
Note [see Theorem 3.9 in [8]] that the set Ais a C-compact subset of X
if and only if every countable functionally open [in X] cover of Ahas a finite
subcover.
The remaining notation can be found in [2].
3. Topological-algebraic properties of function spaces
Interest in studying the C-compact topology generated by a Theorem
3.3 in [7] which characterizes some topological-algebraic properties of the
set-open topology. It turns out if Cλ[X] is a paratopological group [TVS ,
locally convex TVS] then the family λconsists of C-compact subsets of X.
Given a family λof non-empty subsets of X, let λ[C] = {A∈λ: for
every C-compact subset Bof the space Xwith B⊂A, the set [B, U ] is open
in Cλ[X] for any open set Uof the space R}.
Let λmdenote the maximal with respect to inclusion family, provided
that Cλm[X] = Cλ[X]. Note that a family λmis unique for each family λ.
A family λof C-compact subsets of Xis said to be hereditary with respect
to C-compact subsets if it satisfies the following condition: whenever A∈λ
and Bis a C-compact [in X] subset of A, then B∈λalso.
We look at the properties of the family λwhich imply that the space
Cλ[X] with the set-open topology is a topological algebra under the usual
operations of addition and multiplication [and multiplication by scalars].
3
The following theorem is a generalization of Theorem 3.3 in [7].
Theorem 3.1. For a space X, the following statements are equivalent.
1. Cλ[X] = Cλ,u[X].
2. Cλ[X]is a paratopological group.
3. Cλ[X]is a topological group.
4. Cλ[X]is a topological vector space.
5. Cλ[X]is a locally convex topological vector space.
6. Cλ[X]is a topological ring.
7. Cλ[X]is a topological algebra.
8. λis a family of C-compact sets and λ=λ[C].
9. λmis a family of C-compact sets and it is hereditary with respect to
C-compact subsets.
Proof. Equivalence of the statements [1], [3], [4], [5] and [8] proved in [7,
Theorem 3.3].
Note that in the proof of Lemmas 3.1 and 3.2 in [7] used only the con-
dition that the space Xis a paratopological space. Thus [2]⇒[8].
[8]⇒[7]. As [8]⇔[4], we only need to show that continuous operation
of multiplication. Really, let βbe a neighborhood filter of zero function in
C[X]. Let W= [A, V ]∈β, where A∈λand Vis an open set of the space R.
Then there is an open set V1such that V1∗V1⊂V. Show that W1= [A, V 1]
such that W1∗W1⊂W. Indeed W1∗W1={f∗g:f∈W1, g ∈W1}=
{f∗g:f[A]⊂V1and g[A]⊂V1}. Clearly that f[x]∗g[x]∈V1∗V1for
each x∈A. Therefore [f∗g][A]⊂Vand W1∗W1⊂W.
It remains to prove that if W= [A, V ]∈βand f∈C[X] then there is
an open set V1∋0 such that f[A]∗V1⊂Vand V1∗f[A]⊂V. Indeed
let g=f∗hand g1=h1∗fwhere h, h1∈W1. Then g[x] = f[x]∗h[x]∈
f[A]∗V1and g1[x] = h1[x]∗f[x]∈V1∗f[A] for each x∈A. Note that
g[A]⊂Vand g1[A]⊂V.
[8]⇒[9]. Since Cλm[X] = Cλ[X] then Cλm[X] is a topological group
and λmis a family of C-compact sets and consequently, λm=λm[C]. But
if the set [B, U] is open in Cλm[X] for any open set Uof the space Rthen
B∈λm.
Remaining implications is obviously and follow from Theorem 3.3 in [7]
and the definitions.
4
4. Comparison of topologies
In this section, we compare the C-compact-open topology with several
well-known and lesser known topologies.
We use the following notations to denote the particular families of C-
compact subsets of X.
F[X] — the collection of all finite subsets of X.
MC[X] — the collection of all metrizable compact subsets of X.
K[X] — the collection of all compact subsets of X.
SC [X] — the collection of all sequential compact subsets of X.
CC[X] — the collection of all countable compact subsets of X.
P S[X] — the collection of all pseudocompact subsets of X.
RC[X] — the collection of all C-compact subsets of X.
Note that F[X]⊆MC[X]⊆K[X]⊆CC [X]⊆P S [X]⊆RC[X]
and MC[X]⊆SC[X]⊆CC[X]. When λ=F[X], MC[X], K[X],
SC [X], CC[X], P S[X] or RC [X], we call the corresponding λ-open topolo-
gies on C[X] point-open, metrizable compact-open, compact-open, sequen-
tial compact-open, countable compact-open, pseudocompact-open and C-
compact-open respectively. The corresponding spaces are denoted by Cp[X],
Ck[X], Cc[X], Csc[X], Ccc [X], Cps[X] and Crc [X] respectively.
For the C-compact-open topology on C[X], we take as subbase, the fam-
ily {[A, V ] : A∈RC[X], V is open in R}; and we denote the corresponding
space by Crc[X].
We obtain from Theorem 3.1 the following result.
Theorem 4.1. For any space Xand λ∈ {F[X],MC[X],K[X],SC[X],
CC[X],P S [X],RC[X]}, the λ-open topology on C[X]is same as the
topology of uniform convergence on elements of a family λ, that is, Cλ[X] =
Cλ,u[X]. Moreover, Cλ[X]is a Hausdorff locally convex topological vector
space [TVS].
When C[X] is equipped with the topology of uniform convergence on X,
we denote the corresponding space by Cu[X].
Theorem 4.2. For any space X,
Cp[X]≤Ck[X]≤Cc[X]≤Ccc[X]≤Cps [X]≤Crc[X]≤Cu[X]
and
Ck[X][X]≤Csc[X]≤Ccc [X].
5
Now we determine when these inequalities are equalities and give exam-
ples to illustrate the differences.
Example 4.3. Let Xbe the set of all countable ordinals {α:α < ω1}
equipped with the order topology. The space Xis sequential compact and
collectionwise normal, but not compact. For this space X, we have Csc [X]>
Cc[X].
Really, let f=f0and U= [−1,1]. Consider the neighborhood [X, U ]
of f. Assume that there are a family of neighborhoods {[Ai, Ui]}n
i=1, where
Ai— compact, and f∈
n
T
i=1
[Ai, Ui]⊂[X, U ]. Then ∃α < ω1such that
∀β∈Aiβ < α. Define function g:g[β] = 0 for β6αand g[β] = 1 for
β > α. Then g∈
n
T
i=1
[Ai, Ui], but g /∈[X, U ], a contradiction.
Note that for this space X, we have:
Cp[X]< Ck[X] = Cc[X]< Csc[X] = Ccc[X] = Cps[X] = Cr c[X] =
Cu[X].
Example 4.4. Let Y=βNbe Stone-Cˇech compactification of natural num-
bers N. Note that every sequential compact subset of βNis finite. For this
space Y, we have:
Cp[Y] = Ck[Y] = Csc[Y]< Cc[Y] = Ccc [Y] = Cps[Y] = Crc[Y] =
Cu[Y].
Example 4.5. Let Z=X⊕Ywhere Xis the space of Example 4.3 and
Yis the space of Example 4.4. Then the sequential compact-open topology is
incomparable with the compact-open topology on the space C[Z].
Example 4.6. Let X=Icbe the Tychonoff cube of weight c. A space Xis
compact and contains a dense sequential compact subset. Thus, we have:
Cp[X]< Ck[X]< Cc[X] = Csc[X] = Ccc[X] = Cps[X] = Cr c[X] =
Cu[X].
Example 4.7. Let X=ω1+ 1 be the set of all ordinals 6ω1equipped with
the order topology. The space Xis compact and sequentially compact but not
metrizable. Then, for space Xwe have:
Cp[X]< Ck[X]< Csc[X] = Cc[X] = Ccc[X] = Cps[X] = Cr c[X] =
Cu[X].
The following example is an example of the space in which every sequen-
tially compact and compact subset is finite.
6
Example 4.8. Let K0=N. By using transfinite induction, we construct
subspace of βN. Suppose that Kβ⊂βNis defined for each β 0. We claim that X=S∞
n=1 An.
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Suppose that x0∈X\S∞
n=1 An. So there exists a continuous function
f:X7→ [0,1] such that f[x] = 0 for all x∈S∞
n=1 Anand f[x0] = 1.
Since f[x] = 0 for all x∈An,f∈ h0, An, εifor all nand hence, f∈
T∞
n=1h0, An, εi={0}. This means f[x] = 0 for all x∈X. But f[x0] = 1.
Because of this contradiction, we conclude that X is almost σ-C-compact.
[5] ⇒[1]. By Theorem 4.10 in [4] and Theorem 5.1.
By Remark [1] ⇒[6] ⇒[7] ⇒[4].
Corollary 5.5. Suppose that Xis almost σ-C-compact. If Kis a subset of
Crc[X], then the following are equivalent.
1. Kis metrizable compact.
2. Kis compact.
3. Kis sequentially compact.
4. Kis countable compact.
5. Kis pseudocompact.
6. Kis C-compact subset of Crc [X].
A space Xis said to be of [pointwise] countable type if each [point]
compact set is contained in a compact set having countable character.
A space Xis a q-space if for each point x∈X, there exists a sequence
{Un:n∈N}of neighborhoods of xsuch that if xn∈Unfor each n, then
{xn:n∈N}has a cluster point. Another property stronger than being a
q-space is that of being an M-space, which can be characterized as a space
that can be mapped onto a metric space by a quasi-perfect map [a continuous
closed map in which inverse images of points are countably compact]. Both
a space of pointwise countable type and an M-space are q-spaces.
Theorem 5.6. For any space X, the following are equivalent.
1. Crc[X]is metrizable.
2. Crc[X]is of first countable.
3. Crc[X]is of countable type.
4. Crc[X]is of pointwise countable type.
5. Crc[X]has a dense subspace of pointwise countable type.
6. Crc[X]is an M-space.
7. Crc[X]is a q-space.
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8. Xis hemi-C-compact; that is, there exists a sequence of C-compact sets
{An}in Xsuch that for any C-compact subset Aof X,A⊆Anholds
for some n.
Proof. From the earlier discussions, we have [1] ⇒[3] ⇒[4] ⇒[7], [1] ⇒
[6] ⇒[7], and [1] ⇒[2] ⇒[7].
[4] ⇔[5]. It can be easily verified that if Dis a dense subset of a space
Xand Ais a compact subset of D, then Ahas countable character in Dif
and only if Ais of countable character in X. Now since Crc[X] is a locally
convex space, it is homogeneous. If we combine this fact with the previous
observation, we have [4] ⇔[5].
[7] ⇒[8]. Suppose that Crc [X] is a q-space. Hence, there exists a
sequence {Un:n∈N}of neighborhoods of the zero-function 0 in Crc[X]
such that if fn∈Unfor each n, then {fn:n∈N}has a cluster point in
Crc[X]. Now for each n, there exists a closed C-compact subset Anof X
and ǫn>0 such that 0 ∈ h0, An, ǫni ⊆ Un.
Let Abe a C-compact subset of X. If possible, suppose that Ais not a
subset of Anfor any n∈N. Then for each n∈N, there exists an∈A\An. So
for each n∈N, there exists a continuous function fn:X7→ [0,1] such that
fn[an] = nand fn[x] = 0 for all x∈An. It is clear that fn∈ h0, An, ǫni. But
the sequence {fn}n∈Ndoes not have a cluster point in Crc [X]. If possible,
suppose that this sequence has a cluster point fin Crc[X]. Then for each
k∈N, there exists a positive integer nk> k such that fnk∈ hf, A, 1i. So
for all k∈N,f[ank]> fnk[ank]−1 = nk−1>k. But this means that
fis unbounded on the C-compact set A. So the sequence {fn}n∈Ncannot
have a cluster point in Crc[X] and consequently, Crc[X] fails to be a q-space.
Hence, X must be hemi-C-compact.
[8] ⇒[1]. Here we need the well-known result which says that if the
topology of a locally convex Hausdorff space is generated by a countable
family of seminorms, then it is metrizable. Now the locally convex topology
on C[X] generated by the countable family of seminorms {pAn:n∈N}is
metrizable and weaker than the C-compact-open topology. However, since
for each C-compact set Ain X, there exists Ansuch that A⊆An, the locally
convex topology generated by the family of seminorms {pA:A∈RC[X]},
that is, the C-compact-open topology, is weaker than the topology generated
by the family of seminorms {pAn:n∈N}. Hence, Crc[X] is metrizable.
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6. Separable and second countability
Theorem 6.1. For any space Xand λ∈ {MC[X],SC[X],CC[X],P S [X],
RC[X]}, the following are equivalent.
1. Cp[X]is separable.
2. Cc[X]is separable.
3. Xhas a weaker separable metrizable topology.
4. Cλ[X]is separable.
Proof. First, note by Corollary 4.2.2 in [6] that [1], [2], and [3] are equiv-
alent. Also, since Cp[X]6Cλ[X], for λ∈ {MC[X], SC[X], CC [X],
P S[X], RC[X]}, [4] ⇒[1].
[3] ⇒[4]. If Xhas a weaker separable metrizable topology, then Xis
submetrizable. By Theorem 4.14, Ck[X] = Cc[X] = Csc[X] = Ccc [X] =
Cps[X] = Crc [X]. Since [3] ⇒[2], Cλ[X] is separable for each λ∈
{MC[X], SC[X], CC[X], P S[X], RC[X]}.
Corollary 6.2. If Xis pseudocompact and λ∈ {M C [X], K[X], SC[X],
CC[X], P S [X], RC[X]}, then the following statements are equivalent.
1. Cλ[X] is separable.
2. Cλ[X] has ccc.
3. Xis metrizable.
Proof. [1] ⇒[2]. This is immediate.
[2] ⇒[3]. By Corollary 4.8 in [7], Xis metrizable.
[3] ⇒[1]. If Xis metrizable, then X, being pseudocompact, is also
compact. Hence Xis separable and consequenly by Theorem 6.1, Cλ[X] is
separable.
Recall that a family of nonemty open sets in a space Xis called a π-base
for Xif every nonempty open set in Xcontains a member of this family.
The following Theorems are analogues of Theorem 4.6 and Theorem 4.8
in [5].
Theorem 6.3. For a space Xand λ∈ {MC[X],K[X],SC[X],CC[X],
P S[X],RC[X]}, the following statements are equivalent.
13
1. Cλ[X]contains a dense subspace which has a countable π-base.
2. Cλ[X]has a countable π-base.
3. Cλ[X]is second countable.
4. Xis hemicompact and ℵ0-space.
Theorem 6.4. For a locally compact space Xand λ∈ {MC[X],K[X],
SC [X],CC[X],P S[X],RC [X]}, the following statements are equivalent.
1. Cλ[X]is second countable.
2. Xis hemicompact and submetrizable.
3. Xis Lindel¨of and submetrizable.
4. Xis the union of a countable family of compact metrizable subsets of
X.
5. Xis second countable.
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