Compact-open topology PDF
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 < α and |Kβ|6c. Then for each A∈[Sβ<α Kβ]ωchoose xAsuch that xAis an accumulation point of the set Ain the space βN. Let Kα=Sβ<α Kβ∪ {xA:A∈[Sβ<α Kβ]ω}. The space M=Sα<ω1Kαis countable compact space in which every sequentially compact and compact subset is a finite. Thus, we have: Cp(M) = Ck(M) = Csc(M) = Cc(M)< Ccc (M) = Cps(M) = Crc(M) = Cu(M). Example 4.9. Let Mbe a maximal infinite family of infinite subsets of N such that the intersection of any two members of Mis finite, and let Ψ = NSM, where a subset Uof Ψis defined to be open provided that for any set M∈ M, if M∈Uthen there is a finite subset Fof Msuch that {M}SM\ F⊂U. The space Ψis then a first-countable pseudocompact Tychonov space that is not countably compact. The space Ψis due independently to J. Isbell and S. Mr´owka. Every compact, sequentially compact, countable compact subsets of Ψhas the form Sn i=1({xi} ∪ (xi\Si)) ∪S, where xi∈E,|Si|< ω,|S|< ω. Thus obtain the following relations: Cp(Ψ) < Ck(Ψ) = Csc(Ψ) = Cc(Ψ) = Ccc(Ψ) < Cps(Ψ) = Crc (Ψ) = Cu(Ψ). Example 4.10. Let X=βN⊕(ω1+ 1) ⊕M⊕Ψ, where Mis the space of Example 4.8 and Ψis the space of Example 4.9. We have the following relations: Cp(X)< Ck(X)< Csc(X)< Cc(X)< Ccc (X)< Cps(X) = Crc(X) = Cu(X). Example 4.11. Let G= (ω1)⊕M⊕Ψ⊕Ic, where Mis the space of Example 4.8 and Ψis the space of Example 4.9. We have the following relations: Cp(G)< Ck(G)< Cc(G)< Csc(G)< Ccc (G)< Cps(G) = Crc (G) = Cu(G). Example 4.12. Let Y= [0, ω2]×[0, ω1]\ {(ω2, ω1)}, with the topology τ generated by declaring open each point of [0, ω2)×[0, ω1), together with the sets UP(β) = {(β, γ) : γ∈([0, ω1]\P), where Pis finite and (β, ω1)/∈P} and Vα(β) = {(γ, β) : α < γ 6ω2}. 7
Let A={(ω2, γ) : 0 6γ < ω1}and f∈C(Y). Suppose that f(A)is not a closed set, then there are c∈f(A)\f(A)and sequence {an} ⊂ Asuch that {f(an)} → c. Since an= (ω2, γn), there is αnsuch that f(α, γn) = f(an) for each α > αn. Moreover, there is β, such that f(α, γ) = f(ω2, γ)for each γ∈[0, ω1]and α≥β. Clearly that f(β, ω1) = c. Then there is δ, such that f(β, γ ) = cfor each γ≥δ. It follows that f(ω2, γ) = c, but (ω2, γ)∈Aand c /∈f(A), a contradiction. Thus, set Ais a C-compact subset of the space Y. Let Bbe a nonempty pseudocompact subset of Y. Since {α} × [0, ω1]is a clopen set (functionally open) for each α < ω2,([0, ω2]× {β})TBhas at most a finite number of points for each β6ω1. It follows that Bis a compact subset of Y. As Ais the infinite set and closed and the pseudocompact subsets of Yare compact and have at most a finite intersection with A,Aprovides an example of a C-compact subset which is not contained in any closed pseudocompact subset of Y. Since Yhas infinite compact subsets, for this space we have Cc(Y) = Cps(Y)< Crc(Y). Example 4.13. Let Z=YLG, where Yis the space of Example 4.12 and Gis the space of Example 4.11. We have the following relations: Cp(Z)< Ck(Z)< Cc(Z)< Csc(Z)< Ccc (Z)< Cps(Z)< Crc (Z). Recall that a space Xis called submetrizable if Xadmits a weaker metriz- able topology. Note that for a subset Ain a submetrizable space X, the following are equivalent: (1) Ais metrizable compact, (2) Ais compact, (3) Ais sequential compact, (4) Ais countable compact, (3) Ais pseudocompact, (4) Ais C-compact subset of X. Theorem 4.14. Let Xbe a submetrizable space, then Ck(X) = Cc(X) = Csc(X) = Ccc(X) = Cps(X) = Cr c(X). Similarly to Corollary 3.7 in [3] on the bounded-open topology we have Theorem 4.15. For every space X, 8
1. Cc(X) = Crc(X)iff every closed C-compact subset of Xis compact. 2. Crc(X) = Cu(X)iff Xis pseudocompact. Proof. (1) Note that for a subset Aof X,hf, A, ǫi ⊆ hf, A, εi. So if every closed C-compact subset of Xis compact, then Crc(X)≤Cc(X). Conse- quently, in this case, Crc(X) = Cc(X). Conversely, suppose that Cc(X) = Cr c(X) and let Abe any closed C- compact subset of X. So h0, A, 1iis open in Cc(X) and consequently, there exist a compact subset Kof Xand ε > 0 such that h0, K, εi ⊆ h0, A, 1i. If possible, let x∈A\K. Then there exists a continuous function g:X7→ [0,1] such that g(x) = 1 and g(y) = 0 ∀y∈K. Note that g∈ h0, K, εi \ h0, A, 1i and we arrive at a contradiction. Hence, A⊆Kand consequently, Ais compact. (2) First, suppose that Xis pseudocompact. So for each f∈C(X) and each ε > 0, hf, X, εiis a basic open set in Crc (X) and consequently, Cu(X) = Crc(X). Now let Crc(X) = Cu(X). Since h0, X, 1iis a basic neighborhood of the constant zero function 0 in Cu(X), there exist a C-compact subset Aof X and ε > 0 such that h0, A, εi ⊆ h0, X, 1i. As before, by using the complete regularity of X, it can be shown that we must have X=A. But the closure of a C-compact set is also C-compact set. Hence, Xis pseudocompact. Note that for a closed subset Ain a normal Hausdorff space X, the following are equivalent: (1) Ais countable compact, (2) Ais pseudocompact, (3) Ais C-compact subset of X. Corollary 4.16. For any normal Hausdorff space X,Cc(X) = Crc(X) iff every closed countable compact subset of Xis compact. 5. Submetrizable and metrizable One of the most useful tools in function spaces is the following concept of induced map. If f:X7→ Yis a continuous map, then the induced map of f, denoted by f∗:C(Y)7→ C(X) is defined by f∗(g) = g◦ffor all g∈C(Y). Recall that a map f:X7→ Y, where Xis any nonempty set and Yis a topological space, is called almost onto if f(X) is dense in Y. 9
Theorem 5.1. Let f:X7→ Ybe a continuous map between two spaces X and Y. Then (1) f∗:Crc(Y)7→ Cr c(X)is continuous; (2) f∗:C(Y)7→ C(X)is one-to-one if and only if fis almost onto; (3) if f∗:C(Y)7→ Crc(X)is almost onto, then fis one-to-one. Proof. : (1) Suppose g∈Cr c(Y). Let hf∗(g), A, εibe a basic neighborhood of f∗(g) in Crc(X). Then f∗(hg, f (A), εi)⊆ hf∗(g), A, ǫiand consequently, f∗is continuous. (2) and (3) See Theorem 2.2.6 in [6]. Remark 5.2. (1) If a space Xhas a Gδ-diagonal, that is, if the set {(x, x) : x∈X}is a Gδ-set in the product space X×X, then every point in Xis a Gδ- set. Note that every metrizable space has a zero-set diagonal. Consequently, every submetrizable space has also a zero-set-diagonal. (2) Every compact set in a submetrizable space is a Gδ-set. A space Xis called an E0-space if every point in the space is a Gδ-set. So the submetrizable spaces are E0-spaces. For our next result, we need the following definitions. Definition 5.3. A completely regular Hausdorff space Xis called σ-C- compact if there exists a sequence {An}of C-compact sets in Xsuch that X=S∞ n=1 An. A space Xis said to be almost σ-C-compact if it has a dense σ-C-compact subset. Theorem 5.4. For any space X, the following are equivalent. 1. Crc(X)is submetrizable. 2. Every C-compact subset of Crc (X)is a Gδ-set in Crc (X). 3. Every compact subset of Crc(X)is a Gδ-set in Crc(X). 4. Crc(X)is an E0-space. 5. Xis almost σ-C-compact. 6. Crc(X)has a zero-set-diagonal. 7. Crc(X)has a Gδ-diagonal. Proof. (1) ⇒(2) ⇒(3) ⇒(4) are all immediate. (4) ⇒(5). If Crc(X) is an E0-space, then the constant zero function 0 defined on Xis a Gδ-set. Let {0}=T∞ n=1h0, An, εiwhere each Anis C-compact subset in Xand ε > 0. We claim that X=S∞ n=1 An. 10
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. 11
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. 12
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. References [1] R. Arens, J. Dugundji, Topologies for function spaces, Pacific. J. Math.1, (1951), 5–31. [2] R. Engelking, General Topology, PWN, Warsaw, (1977); Mir, Moscow, (1986). [3] S. Kundu, A.B. Raha, The bounded-open topology and its relatives, Rend. Istit. Mat. Univ. Trieste 27 (1995), 61-77. [4] S. Kundu, P. Garg, The pseudocompact-open topology on C(X), Topol- ogy Proceedings, VOL. 30, (2006), 279-299. [5] S. Kundu, P. Garg, Countability properties of the pseudocompact-open topology on C(X): a comparative study, Rend. Istit. Mat. Univ. Trieste 39 (2007), 421-444. [6] R.A. McCoy, I. Ntantu, Topological Properties of Spaces of Continu- ous Functions, Lecture Notes in Math., 1315, Springer-Verlag, Berlin, (1988). [7] A.V. Osipov, Topological-algebraic properties of function spaces with set-open topologies, Topology and its Applications, 159, issue 3, (2012), 800-805. [8] A.V. Osipov, The Set-Open topology, Top. Proc. 37 (2011), 205-217. 14 |