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On Unitary Cauchy Filters on Topological Monoids

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On Unitary Cauchy Filters on Topological Monoids Topological Algebra and its Applications Research Article • DOI: 102478/taa-2013-0006 • TAA • 2013 • 46-59 On unitary Cauchy filters on topological monoids Abstract ∗ For Hausdorff topological monoids, the concept of a unitary Boris G. Averbukh Cauchy net is a generalization of the concept of a fundamen- tal sequence of reals. We consider properties and applica- Moscow State Forestry University, department of mathematics, tions of such nets and of corresponding filters and prove, in Moscow, Russian Federation particular, that the underlying set of a given monoid, endowed with the family of such filters, forms a Cauchy space whose convergence structure defines a uniform topology. A commu- Received 11 November 2012 tative monoid endowed with the corresponding uniformity is uniform. A distant purpose of the paper is to transfer the clas- Accepted 7 October 2013 sical concepts of a completeness and of a completion into the theory of topological monoids. Keywords topological monoid • Cauchy space • uniformity MSC: 22A15, 54D35, 54E15 © 2013 Boris G. Averbukh, licensee Versita Sp. z o. o. This work is licensed under the Creative Commons Attribution-NonCommercial- NoDerivs license, which means that the text may be used for non-commercial pur- poses, provided credit is given to the author. Introduction In analysis, in functional analysis, in differential equations, one often uses sequences of points of normed linear spaces possessing the property that all differences of their far enough members lie in any preassigned neighborhood of zero. In this paper, we define and study nets in an arbitrary Hausdorff topological monoid X which have a similar property. We call them unitary Cauchy nets or shortly C-nets and the corresponding filters C-filters. The first section contains definitions and some elementary properties of such nets and filters. In order to demonstrate the possibility of their application in the theory of topolological monoids, we prove, in particular, by means of them that if a given monothetic topological monoid, whose identity has a neighborhood with a compact closure, is embeddable into a topological group, then it itself is a topological group. In the next section, we show that the underlying set X of X endowed with the family of C-filters is a Cauchy space. We use some principal ideas of the theory of Cauchy spaces and adduce all necessary definitions to make the paper as self-contained as possible. In the third section, we study the standard convergence structure of this Cauchy space and prove that it defines a T 1 3 2 topology, and there exists a canonical uniformity which is compatible with it. We call this topology and this uniformity unitary ones, and we show later that the unitary topology is the finest one amongst topologies on X for which there exists a homeomorphic embedding of X into a topological space where all C-nets defined in the initial topology of X converge. If X is commutative, then its multiplication is uniformly continuous in the unitary uniformity so that X endowed with this multiplication and this uniformity forms a uniform monoid. For this case, we construct a family of subinvariant pseudometrics on X which defines the unitary uniformity. ∗ E-mail: [email protected] 46 On unitary Cauchy filters on topological monoids 1. The notion of a C- filter A) We begin from basic definitions and notations, which we will use throughout the paper. Let X = (X; m; τ) be a Hausdorff topological monoid with an identity 1. Here m is a multiplication and τ a topology on the set X. In the following, we always shorten m(a; b) to ab. Similarly, sometimes, we write X instead of (X; τ) for the underlying space. Definition 1.1. S {x } X C C U α ∈ A A net = α α∈A in is called a -net (a left, a right -net) if for each neighborhood of 1 there exists 0 0 0 0 α ≥ α α ∈ A xα0 ∈ Uxα U xα0 ∈ xα U xα0 ∈ Uxα α ≥ α and for each 0 there exists 0 such that ( , ) for all 0. Here and later, the line on top denotes the topological closure. A version of this definition without this line will be considered in the third paper of this series. The both versions are equivalent, if the closure of some neighborhood of 1 is compact. Sometimes, the presence of this line allows us to deal with filters having bases consisting of closed sets. This property is important by the construction of completions. X R +; C If is ( 0 +) with the usual topology, then -sequences are exactly all increasing fundamental ones. Because of this line, it is also true for the monoid which arises if we exclude all irrationals belonging to some neighborhood of 0. Definition 1.2. A filter F on X (i.e. in the power set P(X)) is called a C-filter (a left, a right C-filter) on X if the set MU = {x ∈ X : UxU ∈ F} (respectively, LU = {x ∈ X : xU ∈ F}, RU = {x ∈ X : Ux ∈ F}) belongs to F for every neighborhood U of 1. We also write MU (LU , RU ) as MU (F) (respectively, LU (F), RU (F)), if we wish to show its dependence on F. It is evident, every left (right) C-net is a C-net. The inclusions LU ⊂ MU and RU ⊂ MU hold for every filter F. Therefore every left (right) C- filter is a C-filter. If X is commutative, then these concepts coincide. As a rule, we only consider C-filters in the following. Left and right C-filters appear sometimes if we discuss properties of left and right translations. All statements proved for C-filters are also true (with obvious changes) for left and right ones. Proposition 1.3. A net S in X is a C-net if and only if the corresponding filter F(S) is a C-filter. A filter F on X is a C-filter if and only if the corresponding net S(F) is a C-net. (See [2] for definitions of the correspondences S → F(S) and F → S(F).) S {x } C U α ∈ A M F S x α ≥ α Proof. If = α α∈A is a -net, then for each there exists 0 such that U ( ( )) contains all α with 0 and hence belongs to F(S). Conversely, if F(S) is a C-filter and U is an arbitrary neighborhood of 1, then there exists α M F S x α ≥ α Ux U F S α 0 such that U ( ( )) contains all α with 0. The set α belongs to ( ) for these . Therefore, there exists 0 0 α xα α ≥ α 0 such that this set contains all with 0. The second statement follows from the first one because of the equality F = F(S(F)). Example 1.4. Let X be a topological group. Then, right (left) C-filters are Cauchy filters of the right (left) uniformity on X. C-filters are Cauchy filters of the Rölke uniformity with a base consisting of entourages {(x; y) ∈ X × X : y ∈ UxU} where U runs symmetrical neighborhoods of 1. B) We introduce now an equivalence relation on the set of C-filters. Definition 1.5. F F C F ≥ F M F ∈ F U F ≈ F Let 1, 2 be -filters. We write 1 2 if U ( 2) 1 for every neighborhood of 1. We set 1 2 if both F ≥ F F ≥ F 1 2 and 2 1 are true. 47 Boris G. Averbukh Definitions of the relations ≥L and ≈L for left C-filters and of the relations ≥R and ≈R for right C-filters are similar. The relations ≥L and ≈L (respectively, ≥R and ≈R ) imply ≥ and ≈. Let S = {xα }α∈A and T = {yβ }β∈B be C-nets. The condition F(S) ≥ F(T ) means: for each neighborhood U of 1 there α ∈ A α ≥ α β ∈ B y ∈ Ux U β ≥ β exists 0 such that for each 0 there exists 0 such that the inclusion β α holds for each 0. We write S ≥ T in such a situation. X R +; S T F S ≈ F T Let, for example, = ( 0 +) with the usual topology and , be increasing fundamental sequences. Then ( ) ( ) means S and T have equal limits. Because of the line on top in Definitions 1.1 and 1.2, it remains true, if we throw out all irrationals belonging to a neighborhood of 0. 0 0 In the following, we use the next denotation. For given subsets H, H of X, we write H ≺ H, if there exists a 0 neighborhood U of 1 such that UH U ⊂ H. For each neighborhood V of an arbitrary point x ∈ X, there exists its 0 0 neighborhood V with V ≺ V . Lemma 1.6. 0 For arbitrary C-filters F , F , the condition F ≥ F implies M 0 F ⊂ M F for arbitrary neighborhoods U, U of 1 2 1 2 U ( 1) U ( 2) 1 with U0 ≺ U. U U0 V VU0V ⊂ U U0xU0 F Proof. For given , , find a neighborhood of 1 such that . The set belongs to the filter 1 for x ∈ M 0 F y ∈ U0xU0 y ∈ M F V yV ∈ F UxU ∈ F every U ( 1). Therefore, there exists a point such that V ( 2) and 2. Then 2 x ∈ M F and U ( 2). Proposition 1.7. ≥ is a quasi-order relation, and ≈ is an equivalence relation on the set of C- filters. ≥ F F F C F ≥ F F ≥ F Proof. It suffices to show that is transitive. Let 1, 2, 3 be -filters with 1 2, 2 3.
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