Filomat 30:1 (2016), 131–140 DOI 10.2298/FIL1601131B
Published by Faculty of Sciences and Mathematics, University of Niˇs, Serbia Available at: http://www.pmf.ni.ac.rs/filomat
Closure Operators in Semiuniform Convergence Spaces Mehmet Barana , Sumeyye Kulaa , T. M. Barana , M. Qasima a Department
of Mathematics, Erciyes University, Kayseri, Turkey.
Abstract. In this paper, the characterization of closed and strongly closed subobjects of an object in category of semiuniform convergence spaces is given and it is shown that they induce a notion of closure which enjoy the basic properties like idempotency,(weak) hereditariness, and productivity in the category of semiuniform convergence spaces. Furthermore, T1 semiuniform convergence spaces with respect to these two new closure operators are characterized.
1. Introduction Semiuniform convergence spaces which form a strong topological universe, i.e., a cartesian closed and hereditary topological category such that products of quotients are quotients are introduced in [30], [31], [32], and [14]. It is well known, [30] or [32], that the construct Conv of Kent convergence spaces can be bicoreflectively embedded in SUConv of semiuniform convergence spaces, and consequently, each semiuniform convergence spaces has an underlying Kent convergence space, namely its bicoreflective Conv-modification. The strong topological universe SUConv contains all (symmetric) limit spaces as well as uniform convergence spaces [16] as a generalization of Weil’s uniform spaces [36] and thus all (symmetric) topological spaces and all uniform spaces. Since topological and uniform concepts are available in SUConv, it is shown, in [31], that semiuniform convergence spaces are the suitable framework for studying continuity, Cauchy continuity, uniform continuity, completeness, total boundedness, compactness, and connectedness as well as convergence structures in function spaces such as simple convergence, continuous convergence, and uniform convergence. There are other known attempts to embed topological and uniform spaces into a common topological supercategory (e.g. quasiuniform spaces by L. Nach [18], syntopogeneous spaces by A. Cs´asz´ar [17], generalized topological spaces (=super topological spaces) by D.B. Doitchinov [22], merotopopic spaces (=seminearness spaces) by M. Kat´etov [26], and nearness spaces by H. Herrlich [23]) that have not even led to cartesian closed topological categories. Closure operators are one of the principal topics both in Categorical Topology and Categorical Algebra. In a category equipped with a notion of subobject and closure one may pursue topological concepts in a context no longer confined to TOP-like categories.
2010 Mathematics Subject Classification. Primary 54B30, 54D10, 54A05, 54A20, 18B99, 18D15. Keywords. Topological category, semiuniform convergence space, T0 objects, T1 objects. Received: 20 February 2014; Revised: 31 August 2014; Accepted: 09 September 2014 Communicated by Dragan S. Djordjevi´c This research was supported by the Scientific and Technological Research Council of Turkey (TUBTAK) under Grant No: 114F299 and Erciyes University Scientific Research Center (BAP) under Grant No: 5627 Email addresses:
[email protected] (Mehmet Baran),
[email protected] (Sumeyye Kula),
[email protected] (T. M. Baran),
[email protected] (M. Qasim)
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Closure operators have always been one of the main concepts in topology. For example, Herrlich [24] characterized coreflections in the category TOP of topological spaces by means of Kuratowski closure operators finer than the ordinary closure. Hong [25] and Salbany [34] used closure operators to produce epireflective subcategories of TOP. A comprehensive study of closure operators in an (E,M)-category and their relations with subcategories along with a variety of examples both in topology and algebra is presented in [19] and [21]. Dikranjan and Giuli [19] introduced a closure operator of a topological category and used this to characterize the epimorphisms of the full subcategories of the given topological category. Baran, in [2] and [3], introduced the notion of (strong) closedness in set-based topological categories and used these notions in [2], [5], [6], [7], [8], [10] and [11] to generalize each of the notions of compactness, connectedness, Hausdorffness, perfectness, regular, completely regular, and normal objects to arbitrary set-based topological categories. Moreover, it is shown, in [9], [10], and [12] that they form appropriate closure operators in the sense of Dikranjan and Giuli [19] in some well-known topological categories. The aim of this paper is to give the characterization of both closed and strongly closed subobjects of an object in category of semiuniform convergence spaces and to show that they form appropriate closure operators (in the sense of [19]) which enjoy the basic properties like idempotency, (weak) hereditariness, and productivity in in category of semiuniform convergence spaces.
2. Preliminaries Let E and B be any categories. The functor U : E → B is said to be topological or that E is a topological category over B if U is concrete (i.e. faithful and amnestic), has small (i.e. sets) fibers, and for which every U−source has an initial lift or, equivalently, for which each U−sink has a final lift [1] or [29]. Note that a topological functor U : E → B is said to be normalized if constant objects, i.e. , subterminals, have a unique structure. Let E be a topological category and X ∈ E. A is called a subspace of X if the inclusion map i : A → X is an initial lift (i.e., an embedding). A filter α on a set B is said to be proper (improper) iff α does not contain (resp., α contains) the empty set, φ. Let F(B) denote the set of filters on B. Let M ⊂ B and [M] = {A ⊂ B : M ⊂ A} and [x] = [{x}]. Note that α ∪ β is the filter {U ∩ V | U ∈ α, V ∈ β } , α∩β = {U ∪ V | U ∈ α, V ∈ β } , and α × β = [{U × V : U ∈ α, V ∈ β}]. If α, β ∈ F(B × B), then α−1 = {U−1 : U ∈ α}, where U−1 = {(x, y) : (y, x) ∈ U}. If U ◦ V = {(x, y):there exist z ∈ B with (x, z) ∈ V and (z, y) ∈ U} , φ for every U ∈ α and every V ∈ β, then α ◦ β is the filter generated by {U ◦ V : U ∈ α, V ∈ β}. Lemma 2.1. Let σ and δ be proper filters on B × B and f : B → C be a function. Then (i) ( f × f )(σ ∩ δ) = ( f × f )(σ) ∩ ( f × f )(δ) and ( f × f )(α) ∪ ( f × f )(β) ⊂ ( f × f )(α ∪ β). (ii) If σ ⊂ δ, then ( f × f )(σ) ⊂ ( f × f )(δ), and if δ is proper filter on C × C , then δ ⊂ ( f f −1 × f f −1 )(δ), where ( f f −1 × f f −1 )(δ) is the proper filter generated by {( f f −1 × f f −1 )(D) : D ∈ δ}. Definition 2.2. (cf. [30] or [32]) 1. A semiuniform convergence space is a pair (B, =), where B is a set = is a set of filters on B × B such that the following conditions are satisfied: (UC1 ) [x] × [x] belongs to = for each x ∈ B. (UC2 ) β ∈ = whenever α ∈ = and α ⊂ β. (UC3 ) α ∈ = implies α−1 ∈ =. If (B, =) is a semiuniform convergence space, then the elements of = are called uniform filters. 0 0 2. A map f : (B, =) → (B , = ) between semiuniform convergence spaces is called uniformly continuous provided 0 that ( f × f )(α) ∈ = for each α ∈ =, where ( f × f )(α) is the proper filter generated by {( f × f )(D) : D ∈ α}. 3. The consctruct of semiuniform convergence spaces (and uniformly continuous maps) is denoted by SUConv.
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2.2 A source { fi : (B, =) → (Bi , =i ), i ∈ I} in SUConv is an initial lift iff α ∈ = precisely when ( fi × fi )(α) ∈ =i for all i ∈ I (cf. [30], [32] p.33 or [15] p.67). 2.3 An epi sink { fi : (Bi , =i ) → (B, =) in SUConv is a final lift iff α ∈ = implies that there exist i ∈ I and βi ∈ =i such that ( fi × fi )(βi ) ⊂ α (cf. [30] , [15] p.67 or [32] p.263). 2.4 The discrete semiuniform convergent structure =d on B is given by =d = {[φ], [x] × [x] : x ∈ B}. 2.5 The indiscrete semiuniform convergent structure on B is given by = = F(B × B). Note that SUConv is a strong topological universe [30]. Recall, in [32] p.31-32, that a generalized convergence space (in [28] and [35] , it is called a filter convergence space and a convergence space, respectively) is pair (B, q), where B is a set and q ⊂ F(B) × B such that the following are satisfied: C1 ) ([x], x) ∈ q for each x ∈ B, C2 ) (β, x) ∈ q whenever (α, x) ∈ q and β ⊃ α. A generalized convergence space (B, q) is called a Kent convergence space [27] (in [28] p.1374, it is called a local filter convergence space) provided that the following is satisfied: C3 ) (α ∩ [x], x) ∈ q whenever (α, x) ∈ q. 0 0 C4 ) A map f : (B, q) → (B , q ) between generalized convergence spaces is called continuous provided 0 that (( f (α), f (x)) ∈ q for each (α, x) ∈ q. C5 ) The category of generalized (filter) convergence spaces and continuous maps is denoted by GConv in [32] (resp., FCO in [35]) , whereas its full subcategory of Kent ( local filter) convergence spaces is denoted by Conv in [27] (resp., LFCO in [28] . Note that every semiuniform convergence spaces (B, =) has an underlying Kent convergence spaces (B, qγ= ) defined as follows: qγ= = {(α, x) : α ∩ [x] ∈ γ= }, where γ= = {β ∈ F(B) : β × β ∈ =} [32] or [30]. 3. Closed Subsets of Semiuniform Spaces In this section, we characterize the (strongly) closed subsets of a semiuniform convergence space. Let B be set and p ∈ B. Let B ∨p B be the wedge at p [2], i.e., two disjoint copies of B identified at p, or in other words, the pushout of p : 1 → B along itself (where 1 is the terminal object in Set, the category of sets). More precisely, if i1 and i2 : B → B ∨p B denote the inclusion of B as the first and second factor, respectively, then i1 p = i2 p is the pushout diagram. A point x in B ∨p B will be denoted by x1 (x2 ) if x is in the first (resp. second) component of B ∨p B. Note that p1 = p2 . The principal p−axis map, Ap : B ∨p B → B2 is defined by Ap (x1 ) = (x, p) and Ap (x2 ) = (p, x). The skewed p−axis map, Sp : B ∨p B → B2 is defined by Sp (x1 ) = (x, x) and Sp (x2 ) = (p, x). The fold map at p, 5p : B ∨p B → B is given by 5p (xi ) = x for i = 1, 2 [2], [3]. Note that the maps Sp and 5p are the unique maps arising from the above pushout diagram for which Sp i1 = (id, id) : B → B2 , Sp i2 = (p, id) : B → B2 , and 5p i j = id, j = 1, 2, respectively, where, id : B → B is the identity map and p : B → B is the constant map at p [10]. The infinite wedge product ∨∞ p B is formed by taking countably many disjoint copies of B and identifying ∞ ∞ them at the point p. Let B∞ = B × B × ... be the countable cartesian product of B. Define A∞ p : ∨p B → B by ∞ Ap (xi ) = (p, p, ..., p, x, p, ...), where xi is in the i-th component of the infinite wedge and x is in the i-th place ∞ ∞ in (p, p, ..., p, x, p, ...), and 5∞ p : ∨p B −→ B by 5p (xi ) = x for all i ∈ I, [2], [3]. ∞ Note, also, that the map Ap is the unique map arising from the multiple pushout of p : 1 → B for which ∞ A∞ i p j = (p, p, ..., p, id, p, ...) : B → B , where the identity map, id, is in the j-th place [10]. Let U : E → Set be a topological functor, X an object in E with U(X) = B. Let M be a nonempty subset of B. We denote by X/M the final lift of the epi U-sink q : U(X) = B → B/M = (B\M) ∪ {∗}, where q is the epi map that is the identity on B\M and identifying M with a point ∗. Let p be a point in B. Definition 3.1. (cf. [2] or [3]) 1. X is T1 at p iff the initial lift of the U−source {Sp : B ∨p B −→ U(X2 ) = B2 and 5p : B ∨p B −→ UD(B) = B} is discrete, where D is the discrete functor which is a left adjoint to U.
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∞ ∞ ∞ ∞ ∞ 2. {p} is closed iff the initial lift of the U−source {A∞ p : ∨p B −→ U(X ) = B and ∇p : ∨p B −→ UD(B) = B} is discrete. 3. M ⊂ X is closed iff {∗} , the image of M, is closed in X/M or M = ∅. 4. M ⊂ X is strongly closed iff X/M is T1 at {∗} or M = ∅. 5. If B = M = ∅, then we define M to be both closed and strongly closed.
Remark 3.2. 1. In Top, the category of topological spaces, the notion of closedness coincides with the usual ones [2], and M is strongly closed iff M is closed and for each x < M there exist a neighborhood of M missing x. If a topological space is T1 , then the notions of closedness and strong closedness coincide [9]. 2. In general, for an arbitrary topological category, the notions of closedness and strong closedness are independent of each other [3]. Theorem 3.3. Let (B, =) be a semiuniform convergence space and let p ∈ B. (B, =) is T1 at p iff for each x , p, [x] × [p] < = and ([x] × [x]) ∩ ([p] × [p]) < =. Proof. Suppose that (B, =) is T1 at p. If [x]×[p] ∈ = for some x , p, then let α = [x1 ]×[x2 ]. Clearly, (π1 Sp ×π1 Sp )(α) = [x] × [p] ∈ =, (π2 Sp × π2 Sp )(α) = [x] × [x] ∈ =, where πi : B2 → B, i = 1, 2, are the projection maps, and (5p × 5p )([x1 ] × [x2 ]) = [x] × [x] ∈ =d , the discrete semiuniform convergence structure on B. Since (B, =) is T1 at p, we get a contradiction. Hence, [x] × [p] < = for all x , p. If ([x] × [x]) ∩ ([p] × [p]) ∈ = for some x , p, then let α = ([x1 ] × [x1 ]) ∩ ([x2 ] × [x2 ]). By Lemma 2.1 (i), (π1 Sp × π1 Sp )(α) = ([x] × [x]) ∩ ([p] × [p]) ∈ =, (π2 Sp × π2 Sp )(α) = [x] × [x] ∈ =, (5p × 5p )(α) = [x] × [x] ∈ =d , a contradiction since (B, =) is T1 at p. Hence, we must have ([x] × [x]) ∩ ([p] × [p]) < = for all x , p. Conversely, suppose that for each x , p, [x] × [p] < = and ([x] × [x]) ∩ ([p] × [p]) < =. We need to show that (B, =) is T1 at p ,i.e., by 2.2, 2.4, and Definition 3.1, we must show that the semiuniform convergence structure =W on B ∨p B induced by Sp : B ∨p B → U((B2 , =2 )) = B2 and 5p : B ∨p B −→ U((B, =d )) = B is discrete, where =2 and =d are the product semiuniform convergence structure on B2 and the discrete semiuniform convergence structure on B, respectively. Let α be any filter in =W , i.e., (πi Sp × πi Sp )(α) ∈ =, i = 1, 2 and (5p × 5p )(α) ∈ =d . We need to show that α = [xi ] × [xi ] (i = 1, 2) or α = [p] × [p] or α = [φ]. If (5p × 5p )(α) = [p] × [p], then α = [pi ] × [pi ] (i = 1, 2) since (5p )−1 {p} = {pi = (p, p)} (i = 1, 2). If (5p × 5p )(α) = [φ], then α = [φ]. If (5p × 5p )(α) = [x] × [x] for some x ∈ B with x , p, then {x1 , x2 } × {x1 , x2 } ∈ α. Therefore α contains a finite set and so there is some M0 ∈ α such that α = [M0 ]. Clearly, M0 ⊆ {x1 , x2 } × {x1 , x2 } and if i , j, then by the first condition, it can be easily shown that (xi , x j ) < M0 and that M0 = {(x1 , x1 ), (x2 , x2 )} is not possible by the second condition. Hence, we must have α = [xi ] × [xi ] i = 1, 2, and consequently, by Definition 3.1, 2.2, and 2.4, (B, =) is T1 at p. Theorem 3.4. Let (B, =) be a semiuniform convergence space and p ∈ B. p is closed in B iff for each x ∈ B with x , p, the following conditions hold. (1) [x] × [p] < =, (2) ([x] × [x]) ∩ ([p] × [p]) < =, (3) ([x] × [x]) ∩ ([p] × [p]) ∩ ([p] × [x]) ∩ ([x] × [p]) < =, Proof. Suppose that {p} is closed in B. If [x] × [p] ∈ = for some x , p, then let α = [x1 ] × [x2 ]. Note that ∞ ∞ ∞ ∞ ∞ (π1 A∞ p × π1 Ap )(α) = [x] × [p] ∈ =, (π2 Ap × π2 Ap )(α) = [p] × [x] ∈ =, (πi Ap × πi Ap )(α) = [p] × [p] ∈ = for i > 3 ∞ ∞ and (5p × 5p )(α) = [x] × [x] ∈ =d , a contradiction since {p} is closed in B. Hence, [x] × [p] < = for all x , p. If ([x] × [x]) ∩ ([p] × [p]) ∈ = for some x , p, then let α = ([x1 ] × [x1 ]) ∩ ([x2 ] × [x2 ]). By Lemma 2.1(i), ∞ ∞ ∞ (π1 A∞ p × π1 Ap )(α) = ([x] × [x])∩ ([p] × [p]) ∈ =, (π2 Ap × π2 Ap )(α) = ([p] × [p]) ∩ ([x] × [x]) ∈ = and for i > 3 ∞ ∞ ∞ ∞ (πi Ap × πi Ap )(α) = ([p] × [p]) ∈ = and (5p × 5p )(α) = [x] × [x] ∈ =d , a contradiction since {p} is closed in B. Hence, ([x] × [x]) ∩ ([p] × [p]) < = for all x , p. If ([x] × [x]) ∩ ([p] × [p]) ∩ ([p] × [x]) ∩ ([x] × [p]) ∈ = for some x , p, then let α = [{x1 , x2 , ...} × {x1 , x2 , ...}]. ∞ It follows that for all i ∈ I, (πi A∞ p × πi Ap )(α) = ([x] × [x]) ∩ ([p] × [p]) ∩ ([p] × [x]) ∩ ([x] × [p]) ∈ = and ∞ ∞ (5p × 5p )(α) = [x] × [x] ∈ =d , a contradiction since {p} is closed in B.
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Conversely, suppose that for each x ∈ B with x , p, [x] × [p] < =, ([x] × [x]) ∩ ([p] × [p] < =, and ([x] × [x]) ∩ ([p] × [p]) ∩ ([p] × [x]) ∩ ([x] × [p]) < =. We need to show that p is closed in B, i.e., by Definition 3.1, ∞ ∞ ∞ ∞ ∞ 2.2, and 2.4, the semiuniform convergence structure =∞ on ∨∞ p B induced by Ap : ∨p B → U((B , = )) = B W ∞ ∞ ∞ and 5p : ∨p B −→ U((B, =d )) = B is discrete, where = and =d is the product semiuniform convergence structure on B∞ and the discrete semiuniform convergence structure on B, respectively. Let α be any filter in =∞ . Then W ∞ ∞ ∞ (πi A∞ × π A )(α) ∈ =, for each i ∈ I and (5 × 5 )(α) ∈ = . By 2.4, we need to show that α = ([x ] × [x ])(i ∈ I) i p i i d p p p or α = [(p, p, p...)] × [(p, p, p, ...)] or α = [φ] ∞ ∞ −1 If (5∞ p × 5p )(α) = [p] × [p], then α = [(p, p, p...)] × [(p, p, p, ...)] since (5p ) {p} = {pi = (p, p, p, ...)}. ∞ ∞ If (5p × 5p )(α) = [φ], then α = [φ]. ∞ If (5∞ p × 5p )(α) = [x] × [x] for some point x in B with x , p, then α contains either a finite set of the form W = {xi1 , xi2 , ...xin } × {xi1 , xi2 , ...xin } or an infinite set of the form W = {x1 , x2 , ...} × {x1 , x2 , ...}. If W = {xi1 , xi2 , ...xin } × {xi1 , xi2 , ...xin } ∈ α, then α contains a finite set and so there is some M0 ∈ α such that α = [M0 ]. Clearly, M0 ⊆ {xi1 , xi2 , ...xin } × {xi1 , xi2 , ...xin } and if ik , il for k, l = 1, 2, ..., n and k , l, then by the first condition, it can be easily shown that (xik , xil ) < M0 and that M0 = {(xi1 , xi1 ), (xi2 , xi2 ), ..., (xin , xin )} is not possible by the second condition. Hence, α = [M0 ] = [xi ] × [xi ], (i ∈ I). If W = {x1 , x2 , ...} × {x1 , x2 , ...} ∈ α, then α contains an infinite set M0 such that α = [M0 ]. Note that ∞ M0 ⊆ W = {x1 , x2 , ...} × {x1 , x2 , ...}. If M0 = W = {x1 , x2 , ...} × {x1 , x2 , ...}, then for each i ∈ I, (πi A∞ p × πi Ap )(α) = ([x] × [x]) ∩ ([p] × [p]) ∩ ([p] × [x]) ∩ ([x] × [p]) ∈ =. Hence, by the third condition, M0 = W = {x1 , x2 , ...} × {x1 , x2 , ...} ∞ is not possible. If M0 = {xk , xk+1 , ...} × {xm , xm+1 , ...} for k, m > 1 and m ≤ k, then, in particular, (πk A∞ p × πk Ap )(α) = ([x]×[x])∩([p]×[p])∩([p]×[x])∩([x]×[p]) ∈ =. Hence, by the third condition, M0 = {xk , xk+1 , ...}×{xm , xm+1 , ...} is ∞ not possible. If M0 = {(xi , x j ), i, j ∈ I, i , j}, then for each i ∈ I, (πi A∞ p × πi Ap )(α) = ([x] × [p]) ∈ = or ([x] × [p]) ∈ =. Hence, by the first condition, M0 = {(xi , x j ), i, j ∈ I, i , j} is not possible. ∞ If M0 = {(xi , xi ), i ∈ I}, then for each i ∈ I, (πi A∞ p × πi Ap )(α) = ([x] × [x]) ∩ ([p] × [p]) ∈ =. Hence, by the second condition, M0 = {(xi , xi ), i ∈ I} is not possible. If M0 = {(xi , xi ), i ∈ I} ∪ {(xk , xk+1 )} or If M0 = {(xi , xi ), i ∈ I} ∪ {(xk , xk+1 ), (xk+3 , xk )} or If M0 = {(xi , xi ), i ∈ I} ∪ {(xk , xk+1 ), (xk+1 , xk ), (xk+3 , xk+10 )}, k fixed, then ∞ (πk−1 A∞ p × πk−1 Ap )(α) = ([x] × [x]) ∩ ([p] × [p]) ∈ =. Hence, by the second condition, these cases of M0 are not possible. Therefore, if α ∈ =∞ , then α = [xi ] × [xi ] (i ∈ I), α = [(p, p, p...)] × [(p, p, p, ...)] or [φ], i.e., by 2.4, the semiuniform W ∞ ∞ ∞ ∞ ∞ ∞ ∞ convergence structure =∞ on ∨∞ p B induced by Ap : ∨p B → U((B , = )) = B and 5p : ∨p B −→ U((B, =d )) = B W is discrete. Hence, by Definition 3.1, {p} is closed in B. Lemma 3.5. Let (B, =) be a semiuniform convergence space, φ , M ⊂ B, β ∈ = and q : U(X) = B → B/M = (B\M) ∪ {∗} be the epi map that is the identity on B\M and identifying M with a point ∗. For all a, b ∈ B with a < M and b ∈ M (i) β ⊂ [b] × [a] or β ∪ ([M] × [a]) is proper iff (q × q)(β) ⊂ [∗] × [a]. (ii) β ∩ ([M] × [M]) ⊂ ([a] × [a]) ∩ ([M] × [M]) and β ∪ ([M] × [M]) is proper iff (q × q)(β) ⊂ ([a] × [a]) ∩ ([∗] × [∗]). (iii)(q × q)(β) ⊂ ([a] × [a]) ∩ ([∗] × [∗]) ∩ ([∗] × [a]) ∩ ([a] × [∗]) iff the following conditions hold. (I) β ⊂ [b] × [a] or β ∪ ([M] × [a]) is proper, (II) β ⊂ [a] × [b] or β ∪ ([a] × [M]) is proper, (III) β ∩ ([M] × [M]) ⊂ ([a] × [a]) ∩ ([M] × [M]) and β ∪ ([M] × [M]) is proper. ([M] × [a]) is proper for all a, b ∈ B with Proof. (i) Suppose that β ∈ =, φ , M ⊂ B, β ⊂ [b] × [a] or β ∪ [∗] [a] a < M and b ∈ M. We need to show that q × q β ⊂ × . If β ⊂ [b] × [a] , then by Lemma 2.1 (ii), q × q β ⊂ q × q ([b] × [a]) = [∗] × [a] . If β ∪ ([M] × [a]) is proper, then for all V ∈ β , V ∩ (M × {a}) , φ. Let W ∈ q × q β . There exists U ∈ β such (M × {a}) , φ. Hence, (b, a) ∈ U for some that W ⊃ q × q (U) . Since β ∪ ([M] × [a]) is proper, it follows that U ∩ b ∈ M and q × q ((b, a)) = (∗, a) ∈ q × q (U) ⊂ W, and consequently, W ∈ [∗] × [a] and q × q β ⊂ [∗] × [a] . Conversely, suppose that β ∈ =, a < M and q × q β ⊂ [∗] × [a] . If β * [b] × [a] and β ∪ ([M] × [a]) is improper for some b ∈ M, then U ∩ (M × {a}) = φ for some U in β. It follows that (c, a) < U for all c ∈ M and
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q × q ((c, a)) = (∗, a) < q × q (U) ∈ q × q β . Hence, q × q β * [∗] × [a] , a contradiction. Hence, β ⊂ [b] × [a] or β ∪ ([M] × [a]) is proper for all a < M and b ∈ M. (ii) Suppose that β ∩ ([M] × [M]) ⊂ ([a] × [a]) ∩ ([M] × [M]) andβ ∪([M] × [M]) is proper. We need to show that [∗]) [∗] [∗] q × q β ⊂ ([a] × [a]) ∩ ([∗] × . First, we will show that q × q β ⊂ × . If q × q β * [∗] × [∗] , then, there exists W ∈ q × q β such that (∗, ∗) < W. Since W ∈ q × q β , there exists V ∈ β such that q × q (V) ⊂ W. (M × M) , φ and consequently, q × q (V ∩ (M × M)) = (∗, ∗) ∈ [M]) β ∪ ([M] × is proper implies that V ∩ q × q (V) ⊂ W. It follows that (∗, ∗) ∈ W, a contradiction. Therefore, we must have q × q β ⊂ [∗] × [∗]. By Lemma 2.1 (i), q × q β ∩ ([M] × [M]) = q × q β ∩ q × q ([M] × [M])= q × q β ∩ ([∗] × [∗]) = q × q β , and consequently q × q β = q × q (β ∩ ([M] × [M]) ⊂ q × q (([a] × [a]) ∩ ([M] × [M]))= ([a] × [a])∩ ([∗] × [∗]). Conversely, suppose that q × q β ⊂ ([a] × [a]) ∩ ([∗] × [∗]) . We will show that β ∪ ([M] × [M]) is proper. If (W) ∈ q × q β ⊂ [∗]×[∗], β∪([M] × [M]) is improper, then there exists W ∈ β such that W ∩(M × M) = φ. q × q (∗, ∗) ∈ q × q (W) . Hence, there exists x, y ∈ W such that q × q x, y = (∗, ∗) . It follows that and consequently, ([M] × [M]) ⊂ x, y ∈ W ∩ (M × M) , a contradiction. Hence, β ∪ ([M] × [M]) is proper. Now, we show that β ∩ ([a] × [a]) ∩ ([M] × [M]) . If U ∈ β ∩ ([M] × [M]), then M × M ⊂ U ∈ β and q × q (U) ∈ q × q β ⊂ ([a] × [a]) ∩ ([∗] × [∗]) = q × q (([a] × [a]) ∩ ([M] × [M])) . It follows that there exists V ∈ ([a] × [a]) ∩ ([M] × [M]) such that q × q (V) ⊂ q × q (U) . Since V ∈ ([a] × [a]) ∩ ([M] × [M]) , V ∩ (M × M) , φ and V ⊂ V ∪ (M × M) = −1 (q × q)−1 ((q × q)(V)) ⊂ q × q q × q (U) = U ∪ (M × M) = U and consequently, U ∈ ([a] × [a]) ∩ ([M] × [M]). Hence, β ∩ ([M] × [M]) ⊂ ([a] × [a]) ∩ ([M] × [M]) . The proof of (iii) follows easily from the Parts (i) and (ii). Theorem 3.6. Let (B, =) be a semiuniform convergence space. φ , M ⊂ B is closed iff for all a, b ∈ B with a < M, b ∈ M and β ∈ =, the conditions (i), (ii), and (iii) hold. (i) β * [a] × [b] and β ∪ ([a] × [M]) is improper. (ii) β ∩ ([M] × [M]) * ([a] × [a]) ∩ ([M] × [M]) or β ∪ ([M] × [M]) is improper. (iii) β * [b] × [a] and β ∪ ([M] × [a]) is improper, or β * [a] × [b] and β ∪ ([a] × [M]) is proper, or β ∩ ([M] × [M]) * ([a] × [a]) ∩ ([M] × [M]) or β ∪ ([M] × [M]) is improper. 0
Proof. M is closed iff, by Definition 3.1, {∗} is closed in B/M iff, by Theorem 3.4, for each a , ∗ in B/M, [a] × [∗] < = , 0 0 0 ([a]×[a])∩([∗]×[∗]) < = , and ([a]×[a])∩([∗]×[∗])∩([a]×[∗])∩([∗]×[a]) < = , where = is the quotient semiuniform 0 structure on B/M that is induced by the map q : B → B/M iff, by definition of = and 2.3, for each β ∈ = and a < M, (q × q)(β) * ([a] × [∗]), (q × q)(β) * ([a] × [a]) ∩ ([a] × [∗]), and (q × q)(β) * ([a] × [a]) ∩ ([∗] × [∗]) ∩ ([∗] × [a]) ∩ ([a] × [∗]) iff, by Lemma 3.5, for all a, b ∈ B and β ∈ = with a < M, b ∈ M, the conditions (i), (ii), and (iii) hold. Theorem 3.7. Let (B, =) be a semiuniform convergence space. φ , M ⊂ B is strongly closed iff for all a, b ∈ B with a < M, b ∈ M and β ∈ =, the following conditions hold. (i) β * [a] × [b] and β ∪ ([a] × [M]) is improper. (ii) β ∩ ([M] × [M]) * ([a] × [a]) ∩ ([M] × [M]) or β ∪ ([M] × [M]) is improper. 0
Proof. M is strongly closed iff, by Definition 3.1, X/M is T1 at ∗ iff, by Theorem 3.3, for each a , ∗ in B/M, [a]×[∗] < = 0 0 [a]) [∗]) and ([a] × ∩ ([∗] × < = iff, by definition of = and 2.3, for each β ∈ = and a < M, q × q β * [a] × [∗] and q × q β * ([a] × [a]) ∩ ([∗] × [∗]) iff, by Lemma 3.5, for all a, b ∈ B and β ∈ = with a < M, b ∈ M, the conditions hold. Theorem 3.8. Let (B, =) be a semiuniform convergence space and p ∈ B. Then, the followings are equivalent. (i) (B, =) is T1 at p. (ii) {p} is strongly closed. (iii) For each x , p, [x] × [p] < = and ([x] × [x]) ∩ ([p] × [p]) < =.
Proof. It follows from Theorem 3.3, Theorem 3.7, and Definition 3.1.
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Note that the full subcategory SConv of SUConv whose object class consists of all convergence spaces is isomorphic to full subcategory ConvS of Conv (= the category of Kent convergence spaces) whose object class consists of all symmetric Kent convergence spaces ( Theorem 3.2 of [31] p. 473). A functor G : ConvS → SConv is defined by G((B, q)) = (B, =q ), where =q = {β ∈ F(B × B) : ∃x ∈ B, ∃α ∈ F(B) with (α, x) ∈ q such that β ⊃ (α × α)} Let (B, q) be a Kent(local filter) convergence space and M ⊂ B. Define K(M) ={x ∈ B : there exists (α, x) ∈ q such that α ∪ [M] is proper} [19] p.198. If K(M) = M, then M is said to be closed ( we call it as closedness in the usual sense). M ⊂ B is closed in a semiuniform convergence space (B, =), if it is closed in its underlying Kent convergence space (B, q= ), where q= = {(α, x) : α ∩ [x] ∈ γ= }, where γ= = {β ∈ F(B) : β × β ∈ =} [32] or [30]. Let (B, q) be a Kent convergence space and M ⊂ B. Note that M ⊂ B is closed in the usual sense iff for all a ∈ B if a < M, then α ∪ [M] is improper for all (α, x) ∈ q ([9], 2.7(1)). Theorem 3.9. Let (B, =) be a semiuniform convergence space. φ , M ⊂ B is strongly closed iff M is closed in the usual sense. Proof. Suppose that M is strongly closed. We show that M is closed in the usual sense. Suppose that for all a ∈ B with a < M, and for all (α, a) ∈ q. Note that (α × α) ∈ =q ⊂ =, and in particular, by Theorem 3.7 (i), (α × α) ∪ ([a] × [M] is improper, and consequently, α ∪ [M] is improper. Hence, M is closed in the usual sense. Suppose that M is closed in the usual sense. We show that M is strongly closed. Suppose that for all a, b ∈ B with a < M, b ∈ M and β ∈ =. We show that the conditions (i) and (ii) of Theorem 3.7 hold. β ∈ = implies that there exists (α, a) ∈ q such that β ⊃ (α × α). We show that β * [a] × [b]. If β ⊂ [a] × [b], then α ⊂ [a] and α ⊂ [b] since β ⊃ (α × α). It follows that α ∪ [M] is proper since b ∈ M and [M] ⊂ [b], a contradiction to M is being closed in the usual sense. Hence, β * [a] × [b]. Now, we show that (β) ∪ ([a] × [M] is improper. Suppose that it is proper. It follows that (α × α) ∪ ([a] × [M] is proper (since β ⊃ (α × α)}), and consequently, α ∪ [M] is proper, a contradiction. The condition (ii) of Theorem 3.4 follows easily. Hence, M is strongly closed. Remark 3.10. Let (B, =) be a semiuniform convergence space and φ , M ⊂ B. Then, by Theorem 3.6, Theorem 3.7, and Theorem 3.9, if M is closed, then M is both strongly closed and closed in the usual sense.
4. Closure Operators Let E be a set based topological category. Recall, (cf. [19] p.132 or [21] p.25), that a closure operator C of E is an assignment to each subset M of (the underlying set of) any object X of a subset CM of X such that: a) M ⊂ CM; b) CN ⊂ CM whenever N ⊂ M ; c) (continuity condition). For each f : X → Y in E and M subset of Y, C( f −1 (M)) ⊂ f −1 (CM), or equivalently, f (CM) ⊂ C( f (M)). M ⊂ X is called C-closed (C-dense ) in X if CM = M (CM = X), and an E-morphism f : X → Y is called C-closed if f (M) is C-closed in Y for each C-closed M in X [21]. A closure operator C is called idempotent if C(CM) = CM, and it is weakly hereditary if every subobject of any object in E is C- dense in its C-closure [19] or [21]. The discrete closure operator δ is defined by setting δ(M) = M for each X ∈ E and M ⊂ X . The trivial closure operator ∂ is defined by setting ∂(M) = X for each X ∈ E and M ⊂ X [21] p. 39 or [20] p.21. The closure operators of E form a large V complete lattice with W δ and ∂ as bottomVand top elements. For closure W operators C and D the meet C D and the join C D are defined by (C D)(M) = C(M) ∩ D(M) and ( C D)(M) = C(M) ∪ D(M) for each X ∈ E and M ⊂ X . The idempotent hull of a closure operator C is denoted by Cˆ (this is the least idempotent closure operator above C ) [19] p.134. A closureWoperator C is W said to be additive if for each X ∈ E and M and N ⊂ X , C(M N) = C(M) ∪ C(N) = C(M)V C(N) [19] or [21], and it is called hereditary if for each X ∈ E and M and N ⊂ X with M ⊂ N, C(MN ) = N C(M), where
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C(MN ) denotes C-closure of M in N [19] or [21]. More on closure operators can be found in [19], [20] and [21]. We, now, show that the notions of closedness and strong closedness form appropriate closure operators clE and sclE of E, where E is the category SUConv of semiuniform convergence spaces and investigate which of the properties idempotency, (weak) hereditariness, additivity and productivity are enjoyed by each of them. Definition 4.1. Let (B, =) be a semiuniform convergence space and M ⊂ B. The (strong) closure of M is the intersection of all (strongly) closed subsets of B containing M and it is denoted by clE (M) (resp. sclE (M)), where E is the category SUConv. 0
Lemma 4.2. Let f : (A, = ) → (B, =) be in SUConv. (1) If D ⊂ B is strongly closed, so also is f −1 (D). (2) Let (B, =) be a semiuniform convergence space. If N ⊂ B is strongly closed and M ⊂ N is strongly closed, so also is M ⊂ B. 0
Proof. (1). Suppose D ⊂ B is strongly closed and for all a, b ∈ A with a < f −1 (D), b ∈ f −1 (D) and β ∈ = , we need to show that h i (i) β * [a] × [b] and β ∪ [a] × f −1 (D) is improper. h i h i h i h i h i h i (ii) β ∩ f −1 (D) × f −1 (D) * ([a] × [a]) ∩ f −1 (D) × f −1 (D) or β ∪ f −1 (D) × f −1 (D) is improper. Note that f (a), f (b) ∈ B with f (a) < D, f (b) ∈ D, and ( f × f )(β) ∈ =. Since D is strongly closed, by Theorem 3.7, we have (i) ( f × f )(β) * f (a) × f (b) and ( f × f )(β) ∪ f (a) × [D] is improper. (ii) ( f × f )(β) ∩ ([D] × [D]) * f (a) × f (a) ∩ ([D] × [D]) or ( f × f )(β)∪ ([D]h × [D])iis improper. If ( f × f )(β) * f (a) × f (b) , then clearly we get β * [a]×[b]. We show that β∪ [a] × f −1 (D) is improper. Suppose i h that it is proper. By Lemma 2.1, we have ( f × f )(β) ∪ f (a) × [D] ⊂ ( f × f )(β) ∪ f (a) × f f −1 (D) ⊂ ( f × f )(β ∪ ([a] × [ f −1 (D)]), and consequently ( f × f )(β) ∪ f (a) × [D] is proper, a contradiction. Hence, β ∪ ([a] × [ f −1 (D)]) is improper. h i h i If ( f × f )(β) ∩ ([D] × [D]) * f (a) × f (a) ∩ ([D] × [D]), then we get β ∩ f −1 (D) × f −1 (D) * ([a] × [a]) ∩ h i h i h i h i f −1 (D) × f −1 (D) . It remains to show that β ∪ f −1 (D) × f −1 (D) is improper. Suppose that it is proper. By h i h i Lemma 2.1, we have ( f × f )(β)∪([D] × [D]) ⊂ ( f × f )(β)∪ f f −1 (D) × f f −1 (D) ⊂ ( f × f )(β∪([ f −1 (D)]×[ f −1 (D)]), and consequently ( f × f )(β) ∪ ([D] × [D]) is proper, a contradiction. Hence, β ∪ ([ f −1 (D)] × [ f −1 (D)]) is improper. (2). Suppose N ⊂ B and M ⊂ N are (strongly) closed, for all a, b ∈ B with a < M, b ∈ M, and β ∈ =. By Theorem 3.7, we need to show that (i) β * [a] × [b] and β ∪ ([a] × [M]) is improper. (ii) β ∩ ([M] × [M]) * ([a] × [a]) ∩ ([M] × [M]) or β ∪ ([M] × [M]) is improper. Suppose a < N. Since N ⊂ B is strongly closed, by Theorem 3.7, we have (i) β * [a] × [b] and β ∪ ([a] × [N]) is improper. (ii) β ∩ ([N] × [N]) * ([a] × [a]) ∩ ([N] × [N]) or β ∪ ([N] × [N]) is improper. Consequently, we get (i) β * [a] × [b] and β ∪ ([a] × [M]) is improper. (ii) β ∩ ([M] × [M]) * ([a] × [a]) ∩ ([M] × [M]) or β ∪ ([M] × [M]) is improper (since M ⊂ N). 0 Suppose that a ∈ N. Since the inclusion map i : (N, = ) → (B, =) is initial lift and β ∈ =, it follows from 2.2 that 0 −1 −1 (i × i) (β) ∈ = ( note that (i × i) (β) = β ∪ [N] × [N] and β ⊂ (i × i)((i × i)−1 (β)). Since M ⊂ N is strongly closed and a, b ∈ N, a < M, b ∈ M, by Theorem 3.7 , (i) (i × i)−1 (β) * [a] × [b] (and consequently, β * [a] × [b] and (i × i)−1 (β) ∪ ([a] × [M]) = β ∪ ([a] × [M]) is improper. (ii) (i × i)−1 (β) ∩ ([M] × [M]) = β ∩ ([M] × [M]) * ([a] × [a]) ∩ ([M] × [M]) or (i × i)−1 (β) ∪ ([M] × [M]) = β ∪ ([M] × [M]) is improper. Hence, by Theorem 3.7, M ⊂ B is strongly closed (since M ⊂ N). 0
Lemma 4.3. Let f : (A, = ) → (B, =) be in SUConv. (1) If D ⊂ B is closed, so also is f −1 (D).
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(2) Let (B, =) be a semiuniform convergence space. If N ⊂ B is closed and M ⊂ N is closed, so also is M ⊂ B. Proof. It is similar to the proof of the Lemma 4.2 by using Theorem 3.6 instead of Theorem 3.7. Theorem 4.4. Let E be SUConv. Both sclE and clE are idempotent, weakly hereditary, productive, and hereditary closure operators of E . Proof. It follows from Definition 4.1, Lemma 4.2, Lemma 4.3, Exercise 2.D, Theorems 2.3 and 2.4, and Proposition 2.5 of [21]. Let E be a set based topological category and C be a closure operator of E. E0C = { X ∈ E : x ∈ C({y}) and y ∈ C({x}) implies x = y}. E1C = {X ∈ E : ∀x ∈ X, C({x}) = {x}}. For E = Top, the category of topological spaces and C = K, the ordinary closure, we obtain the class of T0 -spaces and T1 -spaces, respectively. Theorem 4.5. Let E be SUConv and (B, =) ∈ E. Then, the followings are equivalent. (1) (B, =) ∈ E1scl . (3) (B, =) is T1 . (4) (B, =) is T0 . (5) for any distinct pair of points x and y in B, [x] × [y] < = and ([x] × [x]) ∩ ([y] × [y]) < =. Proof. It follows from Theorem 3.8 and Theorem 4.4, Theorem 4.6, and Remark 4.7(1) of [13]. Let U : E → Set be a topological functor. Recall that a full and isomorphism-closed subcategory S of E is a) epireflective in E iff it is closed under the formation of products and extremal subobjects (i.e., subspaces); b) quotient-reflective in E iff it is epireflective and is closed under finer structures (i.e., if X ∈ S, Y ∈ E, U(X) = U(Y), and id : Y → X is a E-morphism, then Y ∈ S). Theorem 4.6. Let E = SUConv. The subcategory E1scl is quotient-reflective in E. Proof. It is easy to see that this subcategory is full, isomorphism-closed, closed under formation of subspaces, and closedQunder finer structures. It remains to show thatit is closed under formation of products. Let (Bi , =i ) and B = i∈I Bi . Suppose (Bi , =i ) ∈ E1scl . We show that (B, =) ∈ E1scl , where = is the product structure on B. Suppose there exist x , y in B such that [x] × [y] ∈ = or ([x] × [x]) ∩ ([y] × [y]) ∈ =. It follows that there exists m ∈ I such that xm , ym in Bm and (πm × πm )([x] × [y]) = [xm ] × [ym ] ∈ =m or, by Lemma 2.1, (πm × πm )([x] × [x]) ∩ ([y] × [y])) = ([xm ] × [xm ]) ∩ ([ym ] × [ym ]) ∈ =m , a contradiction. Hence, for any x , y in B, [x] × [y] < = and ([x] × [x]) ∩ ([y] × [y]) < =, i.e., (B, =) ∈ E1scl . Hence, the subcategory E1scl is a quotient-reflective subcategory of E. Let (B, q) be a generalized (filter) convergence space and M ⊂ B. Define K(M) ={x ∈ B : there exists (α, x) ∈ q such that α ∪ [M] is proper} [19] p.198 and K∗ (M) = {x ∈ B : K({x}) ∩ M , ∅} ={x ∈ B : (∃c ∈ M) and ([x], c) ∈ q} [20] p.21. Note that K, the ordinary Kuratowski operator, and its opposite K∗ are closure operators. If K(M) = M, then M is said to be closed ( we call it as closedness in the usual sense). M ⊂ B is closed in a semiuniform convergence space (B, =), if it is closed in its underlying Kent convergence space (B, q= ), where q= = {(α, x) : α ∩ [x] ∈ γ= }, where γ= = {β ∈ F(B) : β × β ∈ =} [32] or [30]. Remark 4.7. (1) Let (B, q) be a Kent (local filter) convergence space and M ⊂ B. W (i) By ([9], 2.6(4)), M is strongly closed iff M is K K∗ - closed (the join of K and K∗ ) . (ii) By ([9], 2.7(1)), M is closed (in the usual sense) iff for x ∈ B if there exists (α, x) ∈ q such that α ∪ [M] is proper, then x ∈ M. V (iii) By ([9], 2.5(1)), M is closed (in our sense) iff M is K K∗ - closed (the meet of K and K∗ ). (iv) By (i), (ii), and (iii), if M is strongly closed, then M is closed (in both senses). (v)By (ii) and (iii), if M is closed (in the usual sense), then M is closed (in our sense).
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(2) Let U : E → Set be a topological functor, X an object in E and p ∈ U(X) be a retract of X, i.e., the initial lift h : 1¯ → X of the U−source p : 1 → U(X) is a retract ([4], Theorem 2.6). Then if X is Ti , then X is Ti at p, i = 0, 1 but the converse of implication is not true, in general. (3) Let (B, q) be a Kent (local filter) convergence space and M ⊂ B. By ([9], Theorem 2.5(2)), if (B, q) is T1 , then all subsets of B are closed (in our sense) and M is strongly closed iff M is closed (in the usual sense). (4) By 3.4 of [10], for E = Top and C = sclE , K, K∗ or C = clE , E1C is the class of T1 -spaces. W ∗ W (5) By 2.2.6 of [2], cl=Top = K and scl=Top = K[ K , the idempotent hull of K K∗ . Furthermore, if a topological space X is T1 , then cl=Top = K = scl=Top . V ∗ V W ∗ (6) For E = LFCO, by Theorem 2.5 of [9], clE = K[ K , the idempotent hull of K K∗ , and sclE = K[ K . In particular, if X is T1 , then clE = δ, the discrete closure operator and sclE = K. (7) For an arbitrary set based topological category E, it is not known, in general, whether clE and sclE are closure operators in the sense of [19] or not. Acknowlegment. We would like to thank the referee for his valuable suggestions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36]
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