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Convergence in Neighbourhood Lattices

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Convergence in Neighbourhood Lattices BULL. AUSTRAL. MATH. SOC. 54AO5, 06A99, 06B99, 54C60 VOL. 40 (1989) [129-145] CONVERGENCE IN NEIGHBOURHOOD LATTICES FRANK P. PROKOP A consideration of the separation properties of pre-neighbourhood lattices, leads to the definition and characterisation of T\ -neighbourhood lattices in terms of the properties of the neighbourhood mapping, independently of points. It is then shown that if net conver- gence is defined in neighbourhood lattices as a consequence of replacing 'point' by 'set' in topological convergence, then the limits of convergent nets are unique. The relation- ship between continuity and convergence is established with the proof of the statement that a residuated function between conditionally complete 1\-neighbourhood lattices is continuous if and only if it preserves the limit of convergent nets. If V(X) denotes the power set of X, then the observation that a filter in a topological space (X, T) is a net in V(X) leads to a discussion of the net convergence of a filter as a special case of net convergence. Particular attention is paid to maximal filters, Frechet filters and to the filter generated by the limit elements of a net. Further, if the 'filter' convergence of a filter F in a topological space (X, T) is given by F —* x, if tj(x) C F, then the relationship between 'filter' convergence and the net convergence of a filter in V(X) is established. Finally, it is proved that, in the neighbourhood system 'lifted' from a topological space to TCP(X)), the continuous image of a filter which converges to a singleton set is a convergent filter with the appropriate image set as the limit. 1. INTRODUCTION Neighbourhood lattices were shown in [20] to have the appropriate structure for generalising the non-convergence aspects of topological spaces. This paper will show that not only can convergence be defined in neighbourhood lattices, but also that limits of convergent nets are unique and that a residuated function between conditionally complete T] -neighbourhood lattices is continuous if and only if it preserves the limit of convergent nets. Further, we will show that in conditionally complete Xi-lattices there is a unified theory of convergence, which encompasses both nets and filters. We will make use of the following notation and conventions: If L is an orthocomplemented lattice, then ' will be used to denote complementation; I will be used to denote an arbitrary indexing set; if P is a poset, V and A will represent the operations of sup and inf; while U and fl will be used to denote cue set theoretic operations of union and intersection, V xa will be used to represent. crGI \/{xa | a G I}, with similar abbreviations used for /\ xa, |J Aa, and f) Aa; 1 aG! agl eg! will represent the greatest element of P and 0 will represent the least element of P; Received 20 October 1988 Copyright Clearance Centre, Inc. Serial-fee code: 0004-9729/89 SA2.00+0.00. 129 Downloaded from https://www.cambridge.org/core. IP address: 170.106.40.139, on 26 Sep 2021 at 02:31:17, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700003580 130 F. P. Prokop [2] T(P) = {F | F is a filter of P}; I(P) = {I | / is an ideal of P}; if x 6 P, then [a;) = {j/ I 3/ G P&y ^ i} and (x\ = {y \ y G P&y ^ x}. In addition, the following definitions, which are discussed fully in [20], will be used. If P is an A-semi-lattice, then a function rj: P —» !F(P) is a filter mapping. If 77 is a filter mapping on P, then g G P is neighbourhood open (or simply open) if 77(5) = \g). Further, if 77 is a filter mapping on P, we will let G = {g \ g G P hrj{g) — [g)}. A filter mapping 77 is a pre-neighbourhood mapping if (Vz, < € P){t G 77(3) <3> (3$r G G)(x < 5 ^ t)). A filter mapping 77 is a neighbourhood mapping if i) G is a V-semi-complete sublattice of L, andii) 0el=j. 77(0) = [0). If (X, T) is a topological space, then 77: V(X) -» .F(7>(X)) defined by 77(^4) = {N: (3<7 € T")(-4 C g C iV)} is the induced neighbourhood mapping on "P(X) and the pair (V(X), 77) is the induced neighbourhood lattice of (X, T). Finally, if X and Y are sets and /: X —•• Y is a function, then the direct image function, /»: T(X) —* V(Y) is given by f*(A) = {/(a) | a G .4}, and the inverse image function, /*: T(Y) -• V(X) is given by f*(B) = {x | x 6 X k f{x) € B}. The proofs of those Lemmas wliich are straight forward computations and wliich follow directly from the corresponding definitions will be omitted. 2. To AND Ti NEIGHBOURHOOD LATTICES We have shown in [20] the specific way in which the neighbourhoods of a set in a topological space are determined by the neighbourhoods of its points. This leads to the question: "To what extent are the separation properties 'inherited' by the induced neighbourhood mapping?" We will begin this section by noting that a topological space (X, T) is a TVspace if and only if the induced neighbourhood mapping, 77, restricted to the points of V(X), is one to one. Hence, being a To-space is properly a property of 'points'. Thus, we have DEFINITION 1: An atomic pre-neighbourhood lattice (L, 77) is a To-lattice if 77, restricted to the atoms of L, is one to one. We will now examine how Tj -topological spaces can be characterised in terms of the properties of the induced neighbourhood mapping 77. We begin by looking at the proof of the standard topological statement that a topological space is T\ if and only if the neighbourhood mapping is one to one. Result 2. Let (X, T) be a topological space, and (V(X), 77) be the induced neigh- bourhood lattice. Then (X, T) is a Tj -space *> 77 is a one to one function. PROOF: Let (X, T) be a Tj-space and let A, B £ V(X). Then A £ B =*• (3a € A)(a £ B) => {a}' £ v(A) => v(A) ^ l{B). Conversely, suppose 77 is 1 - 1 on 1 V(X), and x G X. {*}' ^ X => v({x}') ¥= i(X)- Thus, V{{x}') = {{x}') and {x} is open. u Downloaded from https://www.cambridge.org/core. IP address: 170.106.40.139, on 26 Sep 2021 at 02:31:17, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700003580 [3] Convergence in neighbourhood lattices 131 It is clear that a Tb-space is not a Ti-space if and only if there exists A, B £ V(X), A ^ B, for which v(A) = f){B). Further, being a Tj-space is not a point dependent property. Finally, the proof of Result 2 uses only the pre-neighbourhood properties of V- Thus, we can define a T\ -pre-neighbourhood lattice. DEFINITION 3: A pre-neighbourhood lattice (L, 77) is a Ti-lattice if 77: L -+ T(L) is a one to one function. We will now give an alternative characterisation of Ti-lattices. THEOREM 4. Let (L, 77) be a conditionally complete pre-neighbourhood lattice. Then (L, 77) is a Ti-Iattice «• (Va £ L) I /\ x - a 1 . \*e«»(«) ) PROOF: X £ 77(0) => a ^ x. Thus, /\ x exists in L and /\ x ^ a. Let L *e»((o) *e»Ka) be a Tj-lattice, and let b = /\ x. If 6 > o, then 77(0) D 77(6) (proper). But, g an *€»»(«) open element and g £ t](a) => b ^ gr =j- <j (= 77(6). A contradiction. Conversely, a ^ b and 77(0) = 77(6) => /\ K= /\ j/=>a = 6.A contradiction. U It follows from Theorem 4 that (L, 77) is a Ti-lattice «• (Vo £ X) I /\ ff = a I . \9e»)(a)nG I We can now state as a corollary to Theorem 4, a standard equivalent condition for a topological space to be a Tj-space. The "usual" topological proof of this result uses complementation and the fact that points are closed in a Tj-space. We have shown that the proof depends only on the conditional completeness of V(X), and on the pre-neighbourhood and one to one properties of 77. COROLLARY 4.1. Let (X, T) be a topological space, and let (V(X), 77) be the induced neighbourhood lattice. Then (X, T) is a Tt-space <£> (VJI G V(X)) A= )• PROOF: (X, T) is a Ti -space <» {V{X), 77) is a Tx -lattice. D We will now show that in a T\ -lattice the order relation on pre-neighbourhood images must reflect the order relation in the lattice. THEOREM 5. Let (£, 77) be a conditionally complete pre-neighbourhood lattice. Then {L, 77) is a ^-lattice O (Vo, b £ L)(r)(a) C 77(6) => b < a). PROOF: Let (I, n) be a Tj-lattice, and 77(0) C 77(6). x £ 77(0) => (3g £ 77(0) n G) Downloaded from https://www.cambridge.org/core. IP address: 170.106.40.139, on 26 Sep 2021 at 02:31:17, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700003580 132 F. P. Prokop [4] (a < 9 ^ x). But, g € r)(b) =*> b < x.
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