Some New Intrinsic Topologies on Complete Lattices and the Cartesian Closedness of the Category of Strongly Continuous Lattices

and Applied Analysis 3 such that a ∈⇓ d. Thus d ∈⇑ a ⊆ U. Therefore D ∩ U ̸ = 0, hence U ∈ σ(P). This proves τ(P) ⊆ σ(P). (2) Now assume that P is strongly continuous. Then P is continuous. In a strongly continuous lattice, the relation ≪ and ⇐ are the same by the Theorem 2.5 of [5]. Also by [2] in a continuous lattice, every Scott open set A satisfies the condition A = ↑≪A, it follows that every Scott open set of P is T-open. Now let U be S-open. For any directed set D with ⋁D ∈ U, let I = ⋃ d∈D ⇓ d. Then I is a semiprime ideal such that ⋁I = ⋁D. Thus, as U is S-open, I ∩ U ̸ = 0. Hence D ∩ U ̸ = 0. Therefore U is Scott open. By (1) we have τ(P) = σ(P) = κ(P). Lemma 11. For any complete lattice P, if τ(P) = κ(P) then P is semicontinuous. Proof. Suppose that P is not semicontinuous. Then there exists x ∈ P such that x ≰ ⋁ ⇓ x. Then x ∈ U = P\ ↓ (⋁ ⇓ x)which is obviously S-open. Since τ(P) = κ(P),U isT-open. Thus there exists u ≰ ⋁ ⇓ x such that u ⇐ x by Definition 7, a contradiction. Thus Pmust be semicontinuous. Let P be a complete lattice.The long way-below relation ⊲ on P is defined as follows: for any x, y ∈ P, x⊲y if and only if for any nonempty subset B ⊆ P, y ≤ ⋁B implies that x ≤ z for some z ∈ B. For each x ∈ P, we write β (x) = {y ∈ P : y ⊲ x} . (3) Clearly x⊲y implies x ≪ y. In [12], Raney proved that a complete lattice P is a completely distributive lattice if and only if x = ⋁β(x) for all x ∈ P. It is well known that a complete lattice P is a continuous if and only if the topology σ(P) is a completely distributive lattice [2]. If P is semicontinuous, κ(P) is generally not a completely distributive lattice. In Example 9(1), P is semicontinuous and κ(P) consists of all upper subsets of P. Then (κ(P), ⊇) is a complete lattice and the empty set is the largest element of (κ(P), ⊇), but 0 ̸ =⋁β(0). Thus κ(P) is not a completely distributive lattice. Proposition 12. Let (P, ≤) be a complete lattice in which the relation ⇐ satisfies the interpolation property. Then τ(P) is a completely distributive lattice. Proof. By Raney’s characterization of completely distributive lattices, we need to show thatE ≤ ⋁β(E)hold for allE ∈ τ(P). For any x ∈ E ∈ τ(P), by the definition of τ(P), there exists y ∈ E such that x ∈⇑ y ⊆ E. Since the relation ⇐ satisfies the interpolation property,⇑ y ∈ τ(P). Nowwe claim that ⇑ y⊲E. Let D ⊆ τ(P) such that E ≤ ⋁D. Since y ∈ E and ⋁D = ⋃D, there exists U 0 ∈ D such that y ∈ U 0 . Therefore ⇑ y ≤ U 0 . Hence ⇑ y⊲E. It now follows that E ≤ ⋁{⇑ y : y ∈ E} ≤ ⋁β(E). This completes the proof. Lemma 13. If P is a complete lattice such that τ(P) = κ(P), then P is a continuous lattice. Proof. If τ(P) = κ(P), then P is semicontinuous and τ(P) is a completely distributive lattice by Lemma 11 and Proposition 12. By Lemma 10, τ(P) = σ(P) = κ(P), thus (σ(P), ⊆) is a completely distributive lattice. It then follows from Theorem II-1.13 of [2] that P is continuous. This completes the proof. Lemma 14. Let P be a complete lattice. If τ(P) = κ(P), then P is strongly continuous. Proof. By Lemma 11, P is semicontinuous. Assume that P is not strongly continuous. Then there exists b ∈ P such that b < ⋁ ⇓ b. Thus ⋁ ⇓ b ∈ P\ ↓ b which is obviously Sopen. By Lemma 5 and Definition 6, there exists a ∈ P such that a ⇐ b and a ∈ P\ ↓ b. By Lemma 13, a = ⋁↓≪a ∈ P\ ↓ b ∈ σ(P). Thus there exists x ≪ a such that x ≰ b. By Lemma 10, P is continuous and so ↑≪x ∈ σ(P) for all x ∈ P. Again by Lemma 10 and τ(P) = κ(P), τ(P) = σ(P) = κ(P), so ⇑ (↑≪x) = ↑≪x. Thus b ∈⇑ a ⊆⇑ (↑≪x) = ↑≪x. And then x ≪ b. So x ≤ b, which contradicts the assumption on x.This completes the proof. From Lemmas 10 and 14, we obtain the following new characterization of strongly continuous lattices. Theorem 15. A complete lattice P is strongly continuous if and only if τ(P) = κ(P). 3. lim-inf S Convergence and Strongly Continuous Lattices In [2], the lim-inf convergence is introduced, and it is proved that a complete lattice is continuous if and only if the lim-inf convergence on the lattice is topological. This result was later generalized to continuous dcpos and continuous posets (see [2, 13]). Now we introduce a similar convergence, lim-infS convergence, and show that a complete lattice is strongly continuous if and only if the lim-infS convergence on the lattice is topological. Definition 16. A net (xj)j∈J in a complete lattice P is said to lim-infS converge to an element y ∈ P if there exists a semiprime ideal I of P such that (1) ⋁I ≥ y, and (2) for any m ∈ I, x j ≥ m holds eventually (i.e., there exists k ∈ J such that x j ≥ m for all j ≥ k). In this case we write y ≡ lim-inf S x j . For any complete lattice P, define P = {((x j ) j∈J , x) : (x j ) j∈J is a net with x ≡ lim-inf Sxj} . (4) The class P is called topological if there is a topology T on P such that ((x j ) j∈J , x) ∈ P if and only if the net (x j ) j∈J converges to x with respect to the topology T. 4 Abstract and Applied Analysis As in usual cases, associated withP is a family of sets, which is a topology on P: O (P) ={U ⊆ P : U =↑ U and whenever ((xj) j∈J , x) ∈ P and x ∈ U, so eventually x j ∈ U} .