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On the Characterization of $ M $-Metrizable Spaces

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On the Characterization of $ M $-Metrizable Spaces On the characterization of M-metrizable spaces Olli Hella 27 June 2017 ABSTRACT. We introduced the concept of a metric value set (MVS) in an earlier paper [1] and developed the idea further in [2]. In this paper we study locally M-metrizable spaces and the products of M-metrizable spaces. Finally we prove a characterization of commutatively metrizable topological spaces by means of the bases of a quasiuniformity. Introduction First we recall definitions from [1] and [2]. A metric value set (MVS) is a set M with at least two elements and a binary operation + satisfying the following conditions: (M1) The operation + is associative. (M2) The operation + has a (then unique) neutral element e. Let M∗ = M\{e}. (M3) If m1 + m2 = e, then m1 = m2 = e. ∗ ∗ (M4) For every m1, m2 ∈ M there are m3 ∈ M and m4, m5 ∈ M such that m1 = m3 + m4 and m2 = m3 + m5. If the operation + is also commutative, we say that M is a commutative MVS. The operation + induces the relations E and ⊳ in M defined by setting m1 E m2 if there is m3 ∈ M such that m1 + m3 = m2 and m1 ⊳ m2 if there is ∗ m3 ∈ M such that m1 + m3 = m2. We say that M is atom-free if M is commutative and for every m ∈ M∗ there exists n ∈ M∗ such that n ⊳ m. We say that M is strictly atom-free if M is commutative and for every m ∈ M∗ there exists n ∈ M∗ such that n ⊳ m ⋪ n. A quasimetric function f is a map f: X × X M from the square of a set X to an MVS M that satisfies the following conditions:→ (f1) f(x, z) E f(x, y)+ f(y, z) for every x, y, z ∈ X (triangle inequality). (f2) f(x,x)= e for every x ∈ X. arXiv:1707.00615v1 [math.GN] 28 Jun 2017 A quasimetric function f is a metric function if it also satisfies the condition (f3) f(x, y)= f(y, x) for every x, y ∈ X (symmetry). Let f: X × X M be a quasimetric function. Then the topology Tf in X induced by f is defined→ by the family ∗ Bf(x, m)= {y ∈ X : f(x, y) ⊳ m}, m ∈ M , of open balls as the open neighbourhood base at x for every x ∈ X. We call (X, f) a quasimetric space or an M-space and denote Bf : X P(P(X)), Bf(x) = ∗ {Bf(x, m): m ∈ M }, and call Bf an open neighbourhood→ system. 1 Let (X, T ) be a topological space. If there exists a quasimetric function f: X × X M such that the topology Tf induced by f equals T , then we say that (X, T→) or shortly X is an M-metrizable space. We say that X is MVS- metrizable if there is an MVS M such that X is M-metrizable. Let (M1, +1) and (M2, +2) be two metric value sets and h: M1 M2 a map. We say that h is a metric value set homomorphism or shortly→ a homo- morphism if it satisfies the following two conditions: (H1) For all m1 ∈ M it holds that h(m1)= eM2 m1 = eM1 . (H2) h(m +1 n)= h(m)+2 h(n) for every m, n ∈⇐⇒M1. In this paper we define the concept of locally M-metrizable space in a natural way. Then we show that the topology of a metacompact locally M-metrizable space can be induced by a quasimetric function, when M is atom-free. We also study shortly the topology of product spaces. Finally we show how to characterize the class of commutatively metrizable topological spaces by means of the bases of a quasiuniformity. 1 Metric functions and induced topology Definition 1.1. Let X be a set, T a topology on X and U = {Ui : i ∈ I} a family of subsets of X. We say that U is a generating set for T or that it generates T if T is the coarsest topology containing U. This is equivalent to saying that U is a subbase of T . It is well known that the topology T generated by the subbase U is the collection of unions of finite intersections of sets in U, i.e., the sets of the form [ \ Ujk j∈J k∈Kj where J is an arbitrary index set, Kj a finite index set for all j ∈ J and Ujk ∈ U for every j ∈ J and k ∈ Kj. Let X be a set and let (Yi, Ti) be a topological space and fi : X Yi a map for every i ∈ I. The topology of X induced by the family (fi)i∈I is generated→ by −1 the subbase U = {fi U : i ∈ I, U ∈ Ti} Lemma 1.2. Let X be a set and U a family of subsets of X. Let V ⊂ U be such that every element of U is a union of some sets in V. Then U and V generate the same topology. Recall that a map B: X P(P(X)) with B(x) 6= ∅ for every x ∈ X is a neigh- bourhood system for some→ topology on X if it satisfies the following conditions: (B1) x ∈ U for every U ∈ B(x). (B2) If U, V ∈ B(x), then there exists W ∈ B(x) such that W ⊂ U ∩ V. (B3) For every U ∈ B(x) there exists V ∈ B(x) such that for every y ∈ V there exists W ∈ B(y) such that W ⊂ U. Let B satisfy (B1), (B2) and the following condition: (B3’) For every U ∈ B(x) and y ∈ U there exists V ∈ B(y) such that V ⊂ U. Then we say that it is an open neighbourhood system. Since (B3) follows from (B3’), every open neighbourhood system is also a neighbourhood system. Definition 1.3. We say that two neighbourhood systems B and B ′ on a set X are equivalent if they define the same topology. This is true when for every x ∈ X ′ ′ ′ and Bx ∈ B(x) there exists a Bx ∈ B (x) such that Bx ⊂ Bx, and conversely. 2 Lemma 1.4. Let (Yi, Ti) be a topological space and fi : X Yi a map for every i ∈ I. Let Bi be an open neighbourhood system for Yi for→ every i ∈ I. Then the topology induced by the family (fi)i∈I is generated by the subbase −1 V = {fi V : i ∈ I, yi ∈ Yi,V ∈ Bi(yi)}. Proof. Let T be the topology on X induced by the maps fi. The associated −1 generating set of T is U = {fi U : i ∈ I, U ∈ Ti}. Clearly V ⊂ U. Let −1 U fi U ∈ . As an open set in Yi, U can be expressed as an union Sx∈U Bix, B ( ) −1 = −1 V Bix ∈ i x . Thus fi U Sx∈U fi Bix. According to 1.2, generates the same topology as U, which is the topology induced by the maps fi. Let X be a set, Yi be an Mi-metrizable space and fi : X Yi a map for every i ∈ I. In some cases the topology induced by the family (→fi)i∈I is induced by a quasimetric function. Next we study the case where I is a singleton. Theorem 1.5. Let Y be an M-metrizable space and f1 : Y × Y M a quasi- metric function which induces the topology of Y. Let X be a set,→g: X Y a map from X into Y and f2 : X × X M the map defined by setting → → f2(x1,x2)= f1(g(x1), g(x2)). Then f2 is a quasimetric function and the topology T on X induced by g is the same as the topology Tf2 induced by f2. Proof. First we show that f2 is a quasimetric function. Let x1,x2,x3 ∈ X. Now f2(x1,x3)=f1(g(x1), g(x3)) Ef1(g(x1), g(x2)) + f1(g(x2), g(x3)) = f2(x1,x2)+ f2(x2,x3). Thus (f1) holds. The condition (f2) holds, because f2(x,x)= f1(g(x), g(x)) = e. Thus f2 induces a topology Tf2 in X. ∗ We show that Tf2 ⊂ T . For every x ∈ X and m ∈ M the following holds: Bf2 (x, m)={y ∈ X : f1(g(x), g(y)) ⊳ m} ={y ∈ X : g(y) ∈ Bf1 (g(x), m)} −1 =g Bf1 (g(x), m). −1 ∗ To show that T ⊂ Tf2 we need to prove that {g Bf1 (y, m): y ∈ Y, m ∈ M } which is a generating set of T by 1.4, is included in Tf2 . −1 Let V = Bf1 (y, m) and x ∈ g V. Then g(x) ∈ V and therefore m1 = ∗ f1(y, g(x)) ⊳ m. Let m2 ∈ M be such that m1 + m2 = m. Let z ∈ Bf2 (x, m2). Then f1(g(x), g(z)) = f2(x, z) ⊳ m2. Thus by triangle inequality f1(y, g(z)) ⊳ −1 −1 m1 + m2 = m, and therefore z ∈ g V. Hence Bf2 (x, m2) ⊂ g V, which proves that T ⊂ Tf2 . 1.6. Relative topology. Let X be a set with topology T induced by a quasi- metric function f1 : X×X M. Let A ⊂ X and i: A ֒ X be an inclusion. From the previous result we see→ that f2 : A × A M, f2(a→1,a2)= f1(i(a1),i(a2)) = f1(a1,a2), is a quasimetric function, inducing→ on A a topology Tf2 , which is the 3 same as the topology induced by i, i.e., the relative topology on A. From the definition of f2 we see that f2 = f1|A×A.
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