The Centre of the Spaces of Banach Lattice-Valued Continuous Functions on the Generalized Alexandroff Duplicate

and Applied Analysis 3 where Hα is a Γ-open set, Gγ is a Σ-open set, and Mγ is a Γ-compact set. It is easy to see that {Gγ ×{0}}γ∈Ω is an open cover ofK×{0}, thus there is a finite subcoverGγ1 ×{0}, . . . , Gγn ×{0}. Then, Gγ1 × {0, 1} \Mγ1 × {1} ∪ · · · ∪Gγn × {0, 1} \Mγn × {1} 1.6 misses only finitely many Γ-compact sets Mγ1 × {1}, . . . ,Mγn × {1}. AsMγj j 1, 2, . . . , n is compact, we have thatMγj ×{1} ⊂ ∪Hα ×{1}. So,Mγj ×{1} ⊂ ∪p 1Hpj × {1}. Hence, if we add the corresponding open sets from the cover, then we obtain a finite cover of the entire spaceKΣ,Γ ⊗ {0, 1}. Therefore,KΣ,Γ ⊗ {0, 1} is compact. To show that KΣ,Γ ⊗ {0, 1} is Hausdorff, it is enough to show that k, 0 and k, 1 can be separated. Let V be a Γ-open neighborhood of k such that clΓ V closure of V in KΓ is compact. Then,KΣ,Γ ⊗ {0, 1} \ clΓ V × {1} and V × {1} are the separating open sets of k, 0 and k, 1 , respectively. This completes the proof. If KΣ is a compact Hausdorff space without isolated points and KΓ is a discrete topological space, then C KΣ, E ∩ c0 KΓ, E {0} and CD0 KΣ, E C KΣ, E ⊕ c0 KΓ, E is a Banach lattice under the pointwise ordering and supremum norm of the sums f d, where f ∈ C KΣ, E and d ∈ c0 KΓ, E . We refer to 7–9 for more detailed information on these spaces. In 4 , it is showed that CD0 KΣ, E is isometrically Riesz isomorphic to C A K , E , where A K is the Alexandroff duplicate ofK. We will use this identification in the sequel to characterize the centre of the space CD0 KΣ, E . 2. Main Results Let Σ and Γ be compact Hausdorff and locally compact Hausdorff topologies on K, respectively, such that Σ is coarser than Γ, and let E be a Banach lattice. Then C∗ KΣ, Z E s denotes the set of all norm bounded and continuous functions f from K into Z E such that rαf kα e → rf k e in E for each e ∈ E whenever kα, rα → k, r in KΣ,Γ ⊗ {0, 1}. We consider the vector space C KΣ, Z E s × C∗ KΣ, Z E s equipped with coordinatewise algebraic operations, the order 0 ≤ (f, d) ⇐⇒ 0 ≤ f k e , 0 ≤ f k e d k e for each k ∈ K, 2.1