() @ Applied General Topology c© Universidad Politécnica de Valencia Volume 12, no. 1, 2011 pp. 41-47 A Kuratowski-Mrówka type characterization of fibrewise compactness Clara M. Neira U. Abstract In this paper a Kuratowski-Mrówka type characterization of fibrewise compact topological spaces is presented. 2010 MSC: 55R70, 54D30. Keywords: Fibrewise compactness, Kuratowski-Mrówka characterization, tied filter, tied ultrafilter. 1. Introduction Inspired by the concept that the objective of General Topology is the study of continuous functions, a branch of General Topology, known as Continuous Functions Topology or Fibrewise Topology, was originated. To a great extend, the research in this field has been directed to generalize to fibrewise topological spaces, notions and results classically studied in General Topology. Fibrewise versions of Hausdorffness, compactness, convergence, connectedness, uniform structures, and homotopy theory have been studied using the notions of tied filter, and various other tools (cf. [3, 9, 11]). The Kuratowski-Mrówka characterization of compact spaces, as those spaces X satisfying the condition that the second projection π2 : X × Y −→ Y is a closed map, for each space Y , gave rise to a categorical approach of compactness (cf. [5], [6], [8] and [10], among others). In its turn, this characterization has become a useful tool in General Topol- ogy by giving alternative proofs of classic results on compactness that enhance the understanding of some aspects of General Topology. It is worth mentioning 42 C. M. Neira U. the astounding simplicity of the proof of Tychonoff Theorem obtained, in the finite case, via this characterization. In this work we generalize to fibrewise topological spaces the Kuratowski- Mrówka characterization of compactness. 2. Preliminaries In this work we often resort to the following characterization of a closed map. Let X and Y be two topological spaces. A function f : X −→ Y is closed, if and only if, for each y ∈ Y and each open neighborhood O of the fiber Xy = f −1(y) in X, there exists a neighborhood W of y in Y such that XW = {x ∈ X : f (x) ∈ W } ⊂ O (cf. [9], Proposition 1.8, p 7). A fibrewise topological space is by definition a triplet (E, p, T ), where E and T are topological spaces and p : E −→ T is a continuous function. Let (E, p, T ) be a fibrewise topological space. A filter F over E is a tied filter to a point t ∈ T or a t-filter if the filter p(F) generated over T by the filter base {p(F ) : F ∈ F} converges to the point t. A tied ultrafilter to the point t or a t-ultrafilter, is a maximal t-filter. Following I. M. James, we adopt the next definition. Definition 2.1 ([9]). A fibrewise topological space (E, p, T ) is fibrewise com- pact, if p is a proper map. Remark 2.2 ([2]). Recall that a continuous map p : E −→ T is proper if for every topological space Z, the map p × idZ : E × Z −→ T × Z is closed, or equivalently, if p is a closed map and each fiber is compact. The next characterization of a fibrewise compact topological space is quite useful in what follows. Proposition 2.3 ([9]). A fibrewise topological space (E, p, T ) is fibrewise com- pact, if and only if, for each t ∈ T and each covering O of Et by open subsets of E, there exist a neighborhood W of t and a finite subfamily of O that covers EW . Proposition 2.4. Let (E, p, T ) be a fibrewise topological space. The following assertions are equivalent: (1) (E, p, T ) is fibrewise compact. (2) Each filter over E tied to a point t ∈ T has a cluster point in Et (cf. [11]). (3) Each ultrafilter over E tied to a point t ∈ T converges to a point of Et. A Kuratowski-Mrówka type characterization of fibrewise compactness 43 Proof. (1) =⇒ (2): Suppose that the t-filter F over E has no cluster points. Then for each x ∈ Et, there exist a neighborhood Ox of x in E and an element Fx of F such that Ox ∩ Fx = ∅. Since Et is compact, there exist points x1, ..., xn ∈ Et satisfying Et ⊂ ⋃ i=1,...,n Oxi and since p is closed, there exists an open neighborhood W of t in T , such that EW ⊂ ⋃ i=1,...,n Oxi . Let F = ⋂ i=1,...,n Fxi . If s ∈ p(F ) ∩ W , there exists a ∈ F ∩ EW , such that p(a) = s. Then a ∈ Oxi ∩ Fxi , for some i ∈ {1, ..., n}. This is a contradiction, hence p(F ) ∩ W = ∅ and F is not a filter tied to t. It follows that the t-filter F has at least one cluster point. (2) =⇒ (3): Suppose that U is a t-ultrafilter over E and that x ∈ Et is a cluster point of U. If O ∈ V(x), then O ∈ U, otherwise, {O ∩ U : U ∈ U} would generate a t-filter over E finer than U. (3) =⇒ (1): Let t ∈ T and O be a covering of Et by open subsets of E. Suppose that for each open neighborhood W of t and each finite sub-collection A of O one has that EW r ⋃ A 6= ∅. The collection {EW r ⋃ A : W is an open neighborhood of t, and A ⊂ O is finite} is a base for a t-filter over E which is contained in a t-ultrafilter U over E that, by hypothesis, converges to a point x ∈ Et. Now, there exists O ∈ O such that x ∈ O, then O ∈ U, but also E r O = ET r O ∈ U, which is absurd. Then there exist an open neighborhood W of t and a finite sub-collection of O that covers EW . This means that (E, p, T ) is fibrewise compact. � Example 2.5. The triplet (E, p, S1), where E is open interval (0, 2) of R and p : (0, 2) → S1 is defined by p(x) = (cos 2πx, sin 2πx) is a sheaf of sets in which every fiber is a finite set and consequently compact. Let F be the filter over E generated by the collection of intervals {(0, ǫ) : ǫ > 0}. Then F is a filter tied to the point (1, 0) of S1 that has no cluster points in the fiber over (0, 1). It follows that (E, p, T ) is not fibrewise compact. 44 C. M. Neira U. 3. The Kuratowski-Mrówka characterization of fibrewise compactness We begin the main section of this paper with the following observation. Remark 3.1. Every filter F over a set X determines a topology TF over the set X ⋃ {ω}, where ω /∈ X, as follows: if x 6= ω, the neighborhood filter of x is V(x) = {V ⊂ X ⋃ {ω} : x ∈ V } and the neighborhood filter of ω is V(ω) = {F ⋃ {ω} : F ∈ F}. We denote by XF the topological space (X ⋃ {ω}, TF) (cf. [2]). Let (E, p, T ) be a fibrewise topological space and F be a filter over E tied to a point t ∈ T . The function pF : EF −→ T defined by pF (x) = { p(x) if x ∈ E t if x = ω is continuous. That is, (EF , pF , T ) is a fibrewise topological space. To show this, it suffices to verify the continuity of pF at ω. Let W be an open neighborhood of t in T . Since F is a filter tied to t, one has that W ∈ p(F). Then there exists F ∈ F such that p(F ) ⊂ W , hence pF (F ∪ {ω}) ⊂ W . This completes the proof. Let (E, pE , T ) and (F, pF , T ) be two fibrewise topological spaces. The fiber product E ∨ F of E with F is the set E ∨ F = {(x, y) ∈ E × F : p(x) = q(y)}. Consider E ∨ F with the topology induced by the product topology on E × F . The triplet (E ∨ F, p, T ), where p : E ∨ F −→ T is defined by p(x, y) = pE (x), is a fibrewise topological space. Furthermore, (E ∨ F, p, T ) is the product of (E, pE , T ) and (F, pF , T ) in the category of fibrewise topological spaces and fibrewise continuous functions, that is, those continuous functions ϕ : E −→ F satisfying pF ◦ ϕ = pE . Theorem 3.2 (Kuratowski-Mrówka characterization). The fibrewise topologi- cal space (E, p, T ) is fibrewise compact, if and only if, for each fibrewise topo- logical space (F, q, T ) the projection π2 : E ∨ F −→ F is a closed map. Proof. ⇒ Suppose that (E, p, T ) is a fibrewise compact fibrewise topological space and that (F, q, T ) is an arbitrary fibrewise topological space. Let b ∈ F , q(b) = t, and O be an open neighborhood of π−12 (b) = Et × {b} in E ∨ F . For each x ∈ Et there exist a neighborhood Ax of x in E and a neighborhood Mx of b in F such that Ax ∨Mx ⊂ O. Compactness of Et guarantees the existence of x1, ..., xn ∈ Et, such that Et ⊂ ⋃n i=1 Axi . Since p is closed, there exists an open neighborhood W of t in T such that p−1(W ) ⊂ ⋃n i=1 Axi . Let M = ( ⋂n i=1 Mxi ) ⋂ q−1(W ). If (y, a) ∈ π−12 (M ), then p(y) = q(a) ∈ W , hence y ∈ Axi , for some i ∈ {1, ..., n}. Then (y, a) ∈ Axi ∨Mxi ⊂ π −1 2 (b). This proves that π2 is a closed map. A Kuratowski-Mrówka type characterization of fibrewise compactness 45 ⇐ Suppose that F is a filter over E tied to the point t ∈ T and suppose that F has no cluster points, then for each x ∈ Et there exists an open neighborhood Ox of x in E and an element Fx ∈ F such that Ox ⋂ Fx = ∅. Consider the fibrewise topological space (EF , pF , T ) and the set ∆0 = {(x, x) ∈ E ∨ EF : x ∈ E}. For each x ∈ Et, the set Ox ∨ (Fx ⋃ {ω}) is a neighborhood of (x, ω) in E ∨ EF such that Ox ∨ (Fx ⋃ {ω}) ⋂ ∆0 = ∅, then (x, ω) /∈ ∆0 for each x ∈ Et. This implies that π2(∆0) = E and since E is not a closed subset of EF , because ω ∈ E, it follows that π2 : E ∨ EF −→ EF is not a closed map. � Example 3.3. Every topological space X can be identified with the fibrewise topological space (X, p, T ), where T consists of a single point and p is the constant map from E to T . The Kuratowski-Mrówka characterization of the fibrewise compact fibrewise topological spaces asserts that (X, p, T ) is fibre- wise compact if and only if π2 : X ∨ Y −→ Y is closed, for each fibrewise topological space (Y, q, T ). Unfolding this assertion one finds that every fi- brewise topological space (Y, q, T ) can be identified with the topological space Y : the map q is necessarily the constant map from Y to T . Furthermore, X ∨ Y = {(x, y) ∈ X × Y : p(x) = q(y)} = X × Y . Then “A topological space X is compact if and only if, for each topo- logical space Y , each y ∈ Y and each open neighborhood O of X × {y} in X × Y , there exists an open neighborhood N of y in Y , such that X × N ⊂ O.” This result is known in General Topology as the Tube’s Lemma. The second part of the proof of Theorem 3.2 implies the following result. Corollary 3.4. If (E, p, T ) is a fibrewise topological space such that the pro- jection π2 : E ∨ EF −→ EF is closed for every t-filter F over E, then (E, p, T ) is fibrewise compact. Example 3.5. Let (E, p, T ) be a covering space. Since each fiber has the discrete topology, for (E, p, T ) to be fibrewise compact it is necessary, for the fibers, to be finite. Conversely, suppose that (E, p, T ) is a covering space in which every fiber has a finite number of elements. Let F be a filter tied to the point t ∈ T and let Et = {x1, .., xn}. Consider an open neighborhood W of t regularly covered by p and let {Oi}i=1,..., n be a partition in slices of EW with xi ∈ Oi, for each i = 1, ..., n. To guarantee that the function π2 : E ∨ EF −→ EF is closed, consider ζ ∈ EF and an open neighborhood O of π −1 2 (ζ). If ζ ∈ E, V = {ζ} is an open neighborhood of ζ such that π−12 (V ) ⊂ O. 46 C. M. Neira U. Suppose that ζ = ω. For each i = 1, ..., n, there exist an open neighborhood Ai of xi in E and Fi ∈ F in such a way that Ai ∨ (F ∪ {ω}) ⊂ O. Let V = ⋂n i=1 p(Ai). Since V ∈ p(F), there exists F ′ ∈ F such that p(F ′) ⊂ V . Consider F = F ′ ∩ F1 ∩ ... ∩ Fn. If (x, y) ∈ π −1 2 (F ∪ {ω}) and y 6= ω, then p(x) = p(y) ∈ V , therefore x ∈ Ai, for some i = 1, ..., n. Hence (x, y) ∈ Ai ∨ (Fi ∪ {ω}) ⊂ O. This shows that que π2 : E ∨ EF −→ EF is a closed map. Example 3.6. If (E, p, T ) is a fiber bundle with fiber H and if H is compact, then (E, p, T ) is fibrewise compact. In fact, let F be a filter tied to the point t ∈ T . There exist an open neighborhood W of t and a homeomorphism ϕ : p−1(W ) −→ W × H such that π1ϕ = p. To prove that the function π2 : E ∨ EF −→ EF is closed, consider the following facts. (1) Since F is a filter tied to t, then p−1(W ) ∈ F. Therefore, the collection G = {F ∈ F : F ⊂ p−1(W )} is a filter over EW tied to the point t. Here one is considering p−1(W ) as a fibrewise topological space over W . Furthermore, (EW )G is a subspace of EF . (2) Since H is compact, the Kuratowski-Mrówka characterization of com- pact topological spaces guarantees that W ×H, seen as a fibrewise topo- logical space over W , is fibrewise compact. (3) The commutativity of the diagram p−1(W ) ∨ (EW )G (W × H) ∨ (EW )G (EW )G - ϕ×id H H H H H H HHj π2 ? π2 secures that the second projection from p−1(W ) ∨ (EW )G to (EW )G is closed. Let O be an open neighborhood of π−12 (ω) in E ∨ EF . Then O ∩ (p −1(W ) ∨ (EW )G) is a neighborhood of π −1 2 (ω) in p −1(W ) ∨ (EW )G, hence there exists G ∈ G such that π−12 (G ∪ {ω}) ⊂ O ∩ (p −1(W ) ∨ (EW )G). Since G ∈ F, this completes the proof. Remark 3.7. If (E, p, T ) is a fibrewise topological space and U is a t-ultrafilter over E that does not converge, then U has no cluster points. Again, by the second part of the proof of the previous theorem, it follows that π2 : E ∨EU −→ EU is not a closed map. The last observation implies the following corollary. A Kuratowski-Mrówka type characterization of fibrewise compactness 47 Corollary 3.8. If (E, p, T ) is a fibrewise topological space such that the map π2 : E ∨ EU −→ EU is closed for every t-ultrafilter U over E, then (E, p, T ) is fibrewise compact. References [1] J. Adámek, H. Herrlich and G. E. Strecker, Abstract and Concrete Categories. The Joy of Cats, Wiley & Sons, 1990. [2] N. Bourbaki, General Topology, Addison Wesley, 1966. [3] D. Buhagiar, The category MAP, Mem. Fac. 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Willard, General Topology, Addison-Wesley Publishing Company, 1970. (Received July 2010 – Accepted December 2010) Clara M. Neira U. (cmneirau@unal.edu.co) Universidad Nacional de Colombia, Facultad de Ciencias, Departamento de Matemáticas, Carrera 30 # 45–03, Bogotá, Colombia A Kuratowski-Mrówka type characterization of[6pt] fibrewise compactness. By C. M. Neira U.