aa r X i v : . [ m a t h . GN ] F e b ON THE STRUCTURE OF TOPOLOGICAL SPACES
N. MARTINS-FERREIRA
Abstract.
The structure of topological spaces is analysed herethrough the lenses of fibrous preorders. Each topological space hasan associated fibrous preorder and those fibrous preorders whichreturn a topological space are called spacial. A special class ofspacial fibrous preorders consisting of an interconnected family ofpreorders indexed by a unitary magma is called cartesian and stud-ied here. Topological spaces that are obtained from those fibrouspreorders, with a unitary magma I , are called I -cartesian and char-acterized. The characterization reveals a hidden structure of suchspaces. Several other characterizations are obtained and special at-tention is drawn to the case of a monoid equipped with a topology.A wide range of examples is provided, as well as general proceduresto obtain topologies from other data types such as groups and theiractions. Metric spaces and normed spaces are considered as well. February 22, 2021; 01:46:261.
Introduction
The definition of a topological space as we know it today has a longhistory (see e.g. [13]) and it is so rich and full of twists and turnsthat the simple task of tracking down its origins is transformed intoan overwelming undertaking (see e.g. [22]). Arguably, it starts atthe beginning of the twentieth century with seminal works notably byDedekind [8], Lebesgue [17], Riesz [23], de la Vall´e Poussin [7] andFrech´et [9, 10], whose primary interests were still focused on gener-alizing results from the previous century. The first few decades werecharacterized by several further improvements and alternative defini-tions, notably by Hausdorff [11, 12], Carath´eodory [6], Kuratowski [15],Tietze [25], Aleksandrov [1], while the vision of which definitions wouldbe adopted as primitive and which concepts aught to be derived wasnothing but a blur. It was only in the mid thirties that the works
Mathematics Subject Classification.
Primary 06F30, 54H11, 22A15; Sec-ondary 22A05, 17D10.
Key words and phrases. preorder, fibrous preorder, spacial fibrous preorder,cartesian spacial fibrous preorder, topological space, topological group, metricspace, first-countable space, lax-left-associative Mal’tsev operation.This work is supported by the Funda¸c˜ao para a Ciˆencia e a Tecnologia (FCT)and Centro2020 through the Project references: UID/Multi/04044/2019; PAMI -ROTEIRO/0328/2013 (N º by Aleksandrov and Hopf [3], Sierpinski [24], Kuratowski [16] and Lef-schetz [18, 19] started to be influential. The current established notionhas finally settled down with the wide dissemination of the classicalworks by Bourbaki [4] and kelly [14]. From this modern point of view,a topological space is presented as a pair ( X, τ ), where X is a set and τ ⊆ P ( X ) is a topology on X , i.e., a collection of so-called open setswhich are nothing but subsets of X , closed under finite intersectionsand arbitrary unions and moreover at least the empty set and the set X itself must be open.In spite of all the advantages of such an abstract definition, whichhas unquestionably led to great progresses over the last more thanhundred years not only in mathematics but also in physics and in otherareas of knowledge, there are still a few difficulties in the treatment oftopological spaces at this level of abstraction. The trouble is that theprocess of simplifying the definition of topological space to its barebone has two undesirable effects: one is the appearance of redundantinformation, another one is that its true structure is being hidden. Letus mention two concrete examples of this phenomena.If X is a finite set then, as it is well known [2], a topology on X is nothing but a preorder, which is simply a reflexive and transitiverelation. Shouldn’t this be a simple observation that would follow fromthe definition of a topological space?If X is a group and if the map X × X → X ; ( x, y ) xy − is requiredto be continuous (with the product topology on its domain and thegroup operation used to form xy − ) then we get a topological group [21]and the range of possible topologies is severely restricted. Surprisingly,a topological group is presented as a group equipped with an arbitrarytopology (which is required to be compatible with the group operation)while it is clear that such topologies must be simpler than arbitraryones. This is mainly because the simplification on the structure ofthose topologies is not apparent from its definition. Shouldn’t there bea way of identifying which key features of an arbitrary topology givescompatibility with a group operation?Some attempts have been made to overcome these difficulties, no-tably Brown in his book Topology and Groupoids [5]. In [20] the no-tion of spacial fibrous preorder was used to structure the category oftopological spaces so that it becomes apparent that every preorder givesrise to a topology and moreover, for finite sets, there are no possibilitiesother than that.The purpose of this paper is to use some ideas from [20] in a betterunderstanding on the structure of metric spaces and topological groupsfrom the point of view of abstract topological spaces.A spacial fibrous preorder [20] has the advantage of being intuitive(it is in some sense a modification of a preorder) and yet there is a categorical equivalence between spacial fibrous preorders and topolog-ical spaces. Moreover, it suggests that the study of continuous mapsmay be pursued as a fine structure rather than a property. Instead ofa map having the property of being continuous or not it may rather beviewed as a fine structure which can detect different levels of continuitysuch as uniform continuity and other related concepts. Nevertheless,a reader who is not familiar with spacial fibrous preorders in the firstplace may fail to consider it as an appealing structure. For that reasonwe have decided to present here a simpler version which is nonethe-less still sufficient for our purposes. In order to distinguish it from thegeneral case we will call it cartesian spacial fibrous preorder .A fibrous preorder [20] is a sequence R → A → B with R ⊆ A × B satisfying some conditions. If the set A is the cartesian product of aset I and the set B then we will speak of a cartesian fibrous preorder .Not every fibrous preorder is realizable as a topological space, only the spacial ones are so [20]. In the same manner we will restrict cartesianfibrous preorders to spacial ones and simplify its structure a little bitby considering a unitary magma as its indexing set, I , rather thanthe slightly more general structure considered in [20]. Note that when I = { } is a singleton set then A = I × B can be identified with B andthe relation R ⊆ B × B becomes precisely a preorder.2. Cartesian spacial fibrous preorders and unitarymagmas
Let ( I, · ,
1) be a unitary magma. That is a set I together with adistinguished element 1 ∈ I and a binary operation I × I → I ; ( i, j ) i · j, such that i · i = 1 · i for all i ∈ I . Sometimes we write i · j simply as ij . A unitary magma is the same as a monoid when the associativitycondition i ( jk ) = ( ij ) k holds true for all i, j, k ∈ I . Definition 2.1. A cartesian spacial fibrous preorder , indexed by theunitary magma ( I, · , , is a system ( X, ( ≤ i ) i ∈ I , ( ∂ i ) i ∈ I ) where X is aset and for every i ∈ I , ≤ i is a binary relation on X whereas ∂ i is apartial map X × X → I which is defined for all pairs ( x, y ) such that x ≤ i y . Moreover, the following conditions must hold for all x, y, z ∈ X and i, j ∈ I :(C1) x ≤ i x ;(C2) if x ≤ i y , ∂ i ( x, y ) = j and y ≤ j z then x ≤ i z ;(C3) if x ≤ ij y then x ≤ i y and x ≤ j y . A metric space is the best example to illustrate the structure whileproviding useful intuition further on.
Example 2.1.
Let I = N be the unitary magma of natural numberswith ∈ N as the neutral element and the usual multiplication as binary N. MARTINS-FERREIRA operation. Let X be any metric space with metric d : X × X → [0 , + ∞ [ .Under these assumptions we put x ≤ n y if and only if d ( x, y ) < n andchoose ∂ n ( x, y ) = k to be such that k ≤ n − d ( x, y ) , with x, y ∈ X and n, k ∈ N . It is not difficult to see that conditions (C1)-(C3) aresatisfied. As for metric spaces, cartesian spacial fibrous preorders give rise totopological spaces.
Proposition 2.1.
Every cartesian spacial fibrous preorder ( X, ( ≤ i ) i ∈ I , ( ∂ i ) i ∈ I ) gives rise to a topological space ( X, τ ) with τ defined as O ∈ τ ⇔ ∀ x ∈ O , ∃ i ∈ I, N ( i, x ) ⊆ O (1) where N ( i, x ) = { y ∈ X | x ≤ i y } .Proof. This is a special case of the equivalence betwen spacial fibrouspreorders and topological spaces [20]. Having a cartesian spacial fi-brous preorder ( X, ( ≤ i ) i ∈ I , ( ∂ i ) i ∈ I ) we define a spacial fibrous preorder( R, A, B, ∂, p, s, m ) (see [20]) as: R ⊆ I × X × X, A = I × X, B = X (2)with R , ∂ : R → A , p : A → B , s : B → A and m : A × B A → A , de-fined as follows: R = { ( i, x, y ) | x ≤ i y } ,∂ ( i, x, y ) = ( ∂ i ( x, y ) , y ) ,p ( i, x ) = x,s ( x ) = (1 , x ) ,m ( i, j, x ) = ( i · j, x ) . Nevertheless, a direct proof is easily obtained by showing that N ( i, x )is a system of open neighbourhoods. (cid:3) It is clear that when a cartesian spacial fibrous preorder is obtainedfrom a metric space then its induced topology is the same as the usualtopology generated by the metric.3.
A characterization of I-cartesian spaces
The following result characterizes those topological spaces that areobtained from a cartesian spacial fibrous preorder. Such spaces will becalled I -cartesian when I is the indexing unitary magma. Most of thetime we will be considering an arbitrary unitary magma, I , however,it is useful from time to time to recall that our motivating example isthe unitary magma of natural numbers. For that reason we will usethe letters i, j, k to represent elements in the set I as well as the letters n, m, k . Theorem 3.1.
Let ( I, · , be a unitary magma and ( X, τ ) a topologicalspace. The following conditions are equivalent:(a) The space X is I -cartesian.(b) The topology τ is determined by a cartesian spacial fibrous pre-order ( X, ( ≤ i ) i ∈ I , ( ∂ i ) i ∈ I ) as O ∈ τ ⇔ ∀ x ∈ O , ∃ i ∈ I, N ( i, x ) ⊆ O (3) with N ( i, x ) = { y ∈ X | x ≤ i y } .(c) There exists a map N : I × X → P ( X ) such that(i) x ∈ N ( n, x ) , for all n ∈ I , x ∈ X (ii) N ( n, x ) ⊆ { y ∈ X | ∃ k ∈ I, N ( k, x ) ⊆ N ( n, x ) } ,(iii) N ( n · m, x ) ⊆ N ( n, x ) ∩ N ( m, x ) , n, m ∈ I , x ∈ X for which τ is determined as O ∈ τ ⇔ ∀ x ∈ O , ∃ n ∈ I, N ( n, x ) ⊆ O . (d) There exists a ternary relation R ⊆ I × X × X together with amap p : R → I such that(i) ( n, x, x ) ∈ R , for all n ∈ I , x ∈ X (ii) if ( n, x, y ) ∈ R and ( p ( n, x, y ) , y, z ) ∈ R then ( n, x, z ) ∈ R (iii) if ( n · m, x, y ) ∈ R then ( n, x, y ) ∈ R and ( m, x, y ) ∈ R for which τ is determined as O ∈ τ ⇔ ∀ x ∈ O , ∃ n ∈ I, N R ( n, x ) ⊆ O with N R ( n, x ) = { y ∈ X | ( n, x, y ) ∈ R } .(e) There are maps η : I × X → τ and γ : { ( U, x ) | x ∈ U ∈ τ } → I such that:(i) x ∈ η ( n, x ) for every n ∈ I and x ∈ X ,(ii) η ( γ ( U, x ) , x ) ⊆ U , for all x ∈ U ∈ τ ,(iii) η ( n · m, x ) ⊆ η ( n, x ) ∩ η ( m, x ) , for all n, m ∈ I , and x ∈ X .Proof. Conditions ( a ) and ( b ) are equivalent by definition.To prove ( b ) implies ( c ) we start with a cartesian spacial fibrouspreorder and define N ( n, x ) = { y ∈ X | x ≤ n y } . Conditions ( c )( i ) and( c )( iii ) follow respectively from axioms ( C
1) and ( C c )( ii )we start with y ∈ N ( n, x ), that is, x ≤ n y , and observe that there exists k = ∂ n ( x, y ) ∈ I such that N ( k, y ) ⊆ N ( n, x ). Indeed, if z ∈ N ( k, y ),that is, y ≤ k z , then by ( C
2) we have x ≤ n z , which is the sameas saying z ∈ N ( n, x ). This shows that the map N : I × X → P ( X )satisfies ( c )( ii ). It remains to show that τ is determined by it. Thisfollows from Proposition 2.1 and the assumption that τ is determinedas in equation (3).In order to prove ( c ) implies ( d ) we define R = { ( n, x, y ) ∈ I × X × X | y ∈ N ( n, x ) } and put p ( n, x, y ) = k for some k ∈ I such that N ( k, y ) ⊆ N ( n, x ),which exists by assumption on condition ( c )( ii ). Once again, conditions N. MARTINS-FERREIRA ( d )( i ) and ( d )( iii ) are direct consequences of ( c )( i ) and ( c )( iii ), respec-tively. To see that ( d )( ii ) is satisfied we observe that if y ∈ N ( n, x ) and z ∈ N ( p ( n, x, y ) , y ), then, by definition of p ( n, x, y ), we have N ( k, y ) ⊆ N ( n, x ). It follows z ∈ N ( n, x ) and hence ( d )( ii ) is satisfied. Having R we define N R = { y | ( n, x, y ) ∈ R } = { y ∈ N ( n, x ) } = N ( n, x ) andso the topology τ is obtained by N R = N .To prove ( d ) implies ( e ), define η ( i, x ) = { y ∈ X | ( n, x, y ) ∈ R } = N R ( n, x ) and put γ (( U, x )) = k for some k ∈ I such that N R ( k, x ) ⊆ U , which exists by the assumptionthat τ is generated by N R . The map η : I × X → τ is well definedbecause each η ( n, x ) ∈ τ . Indeed, if y ∈ η ( n, x ), that is ( n, x, y ) ∈ R ,then there exists k = p ( n, x, y ) ∈ I for which N R ( k, y ) ⊆ N R ( n, x ).This is a consequence of ( d )( ii ). If z ∈ N R ( k, y ) ⇔ ( k, y, z ) ∈ R then, given that ( n, x, y ) ∈ R and k = p ( n, x, y ), we have ( n, x, y ) ∈ R or, in other words, z ∈ N R ( n, x ). This shows that each η ( n, x ) isopen in τ . Conditions ( e )( i ) and ( e )( iii ) are direct consequence of( d )( i ) and ( d )( iii ), respectively. To show ( e )( ii ) we observe that if y ∈ η ( γ ( U, x ) , x ) then ( γ ( U, x ) , x, y ) ∈ R , but γ ( U, x ) = k , for some k ∈ I such that N R ( k, x ) ⊆ U . Since y ∈ N R ( k, x ) ⇔ ( k, x, y ) ∈ R , weconclude that y ∈ U . This shows that η ( γ ( U, x ) , x ) ⊆ U .Finally, we prove ( e ) implies ( b ). Having η and γ it is not difficultto see that a cartesian spacial fibrous preorder is obtained if we let x ≤ i y ⇔ y ∈ η ( i, x ) ∂ i ( x, y ) = γ ( η ( i, x ) , y )with i ∈ I and x, y ∈ X . Indeed, axioms ( C
1) and ( C
3) follow re-spectively from ( e )( i ) and ( e )( iii ). For ( C
2) let us suppose x ≤ i y ,that is, y ∈ η ( i, x ), and let us suppose y ≤ k z with k = ∂ i ( x, y ) = γ ( η ( i, x, y )). This means z ∈ η ( k, y ) and by condition ( e )( ii ) we have η ( γ ( η ( i, x ) , y ) , y ) ⊆ η ( n, x ), thus we have z ∈ η ( i, x ), or x ≤ i z asdesired.It remains to show that the topology τ is recovered as prescribed in(3). On the one hand if O ∈ τ and if x ∈ O then there is k = γ ( O , x )with N ( k, x ) ⊆ O . This means that every open set in τ is generatedas in condition (3). To see the converse let us consider any subset O ⊆ X and suppose it has the property that for all x ∈ O there issome k = k ( x ) ∈ I with N ( k, x ) = η ( k, x ) ⊆ O . We have to show that O ∈ τ . This follows because O = [ x ∈O N ( k ( x ) , x )and every N ( k, x ) = η ( k, x ) ∈ τ . (cid:3) We immediately observe some interesting special cases, namely when I = { } is the trivial unitary magma, or when I = ( N , · ,
1) is the uni-tary magma (monoid) of natural numbers with the usual multiplica-tion. Furthermore, as we will see in Section 6, the map p : R → I ofcondition ( d ) in the previous result can sometimes be decomposed as p ( n, x, y ) = β ( x, y ) · n in which β ( x, y ) = γ ( η (1 , x ) , y ) with η and γ asin condition ( e ) above.In a sequel to this work our attention will be turned to morphismsbetween fibrous preorders and on how they can be defined internallyto any category with finite limits.For the moment, let us briefly mention that a morphism betweencartesian spacial fibrous preorders, say from( X, ( ≤ i ) i ∈ I , ( ∂ i ) i ∈ I )to ( Y, ( ≤ i ) i ∈ I , ( ∂ i ) i ∈ I ) , consists of a map f : X → Y and a family of maps ( g j : X → I ) j ∈ I suchthat x ≤ g j ( x ) y ⇒ f ( x ) ≤ j f ( y ) (4)for all x, y ∈ X and j ∈ I . Note that when each g j can be chosenindependently from x then the family ( g j ) j ∈ I may be seen as a singlemap g : I → I and this is comparable to the notion of uniform conti-nuity (see the example of metric spaces presented above). It is alsoclear that morphisms between cartesian spacial fibrous preorders maybe considered for different indexing unitary magmas. This and otherconsiderations will be explored in a sequel to this work.So far we have isolated one basic ingredient in the structure of thosetopological spaces which are obtained as cartesian spacial fibrous pre-orders: a unitary magma. Note that every ascending chain on a set I makes it a unitary magma with the smallest element as neutral and asbinary operation the procedure of choosing the greatest between anytwo.We now ask: Besides a unitary magma, is it possible that the struc-ture of a system ( X, ( ≤ i ) i ∈ I , ( ∂ i ) i ∈ I ), inducing a topology on X , canfurther be decomposed into simpler structures which are hence easierto analyse?The answer to this question, as we will see, is yes provided only thatwe restrict its generality a little bit further. Nevertheless, we are stillable to capture the majority of examples of which we are interested in,namely preorders, metric spaces and certain special classes of groupsand monoids with a topology (see Sections 5 and 6). First, let usanalyse the concrete examples. N. MARTINS-FERREIRA Some examples of cartesian spacial fibrous preorders
Let us start with two simple cases of interest as unitary magmas. Thetrivial (singleton) unitary magma, and the unitary magma of naturalnumbers with usual multiplication as its binary operation.It is clear that a cartesian spacial fibrous preorder, indexed by asingleton unitary magma I = { } , is nothing but a preorder. Indeed,in that case we always have ∂ i ( x, y ) = i with i = 1 and hence condi-tion (C2) becomes transitivity. Condition (C1) asserts reflexivity whilecondition (C3) is trivial. Proposition 4.1.
The category of preorders is equivalent to the cate-gory of -cartesian topological spaces which are the same as Aleksandrovor discrete spaces.Proof. Preorders are precisely cartesian spacial fibrous preorders in-dexed by a singleton unitary magma. Moreover, if I = { } then condi-tion (4) asserts the monotonicity of the map f . The result follows fromTheorem 3.1. It is well-known that Aleksandrov spaces, or discretespaces, are the same as preorders [2] (see also [20]). (cid:3) When I = ( N , · ,
1) is the unitary magma of natural numbers withusual multiplication then we have several interesting cases.A natural space [20] is essentially a first-countable topological space.It consists of a pair (
X, N ) with X a set and N : N × X → P ( X ) amap into the power-set of X satisfying the following three conditionsfor all x ∈ X , and i, j ∈ N : x ∈ N ( i, x ) (5) N ( i, x ) ⊆ { y ∈ X | ∃ j ∈ N , N ( j, y ) ⊆ N ( i, x ) } (6) N ( ij, x ) ⊆ N ( i, x ) ∩ N ( j, x ) . (7)Natural spaces are precisely N -cartesian spaces (see item ( c ) of The-orem 3 . X, d ) gives rise to a cartesianspatial fibrous preorder indexed by the natural numbers as alreadypresented. Put x ≤ n y whenever d ( x, y ) < n and define ∂ n ( x, y ) = k such that k ≤ n − d ( x, y ).Similarly, every normed vector space gives rise to a cartesian spatialfibrous preorder indexed by the natural numbers. We simply put x ≤ n y whenever k y − x k < n and define ∂ n ( x, y ) = k ∈ N as any naturalnumber such that k ≤ n − k y − x k . However, if reformulated as: x ≤ n y ⇔ n k y − x k < ∂ n ( x, y ) = rn with r ∈ N any natural number such that for all u ∈ X r k u k < ⇒ k u + n ( y − x ) k < , (9)then we observe that the needed topological information is reduced to: (1) the open ball, B , of radius 1 and centred at the origin;(2) a choice of a natural number r = r ( z ), for every z ∈ B , suchthat for all u ∈ Xr k u k < ⇒ k u + z k < . (10)This is sufficient motivation to consider those topological groups( X, , + , τ ) for which there exists an open neighbourhood of the origin,say B ⊆ X , satisfying the following condition: ∀ x ∈ B, ∃ n ∈ N , B n + x ⊆ B, (11)which, in other words, if defining B n = { u ∈ X | nu ∈ B } , can bestated as ∀ x ∈ B, ∃ n ∈ N , ∀ u ∈ X, nu ∈ B ⇒ u + x ∈ B. (12)We observe, furthermore, that there is no need to start with a topo-logical group. Any group with a subset B , containing the origin andsatisfying the previous condition, immediately satisfies conditions (C1)and (C2) in the definition of a cartesian spacial fibrous preorder as soonas we put x ≤ i y whenever n ( y − x ) ∈ B and define ∂ n ( x, y ) = rn with r ∈ N any natural number such that ∀ u ∈ X, ru ∈ B ⇒ u + n ( y − x ) ∈ B. (13)Indeed, for every n ∈ N , we get x ≤ n x if and only if n ( x − x ) ∈ B or equivalently 0 ∈ B . Now, if x ≤ n y and y ≤ rn z with r ∈ N suchthat (13) holds, then as a consequence we have x ≤ n z . To see itlet us take any n ( y − x ) ∈ B and rn ( z − y ) ∈ B , then, by taking u = n ( z − y ) ∈ X , we observe ru ∈ B and hence u + n ( y − x ) ∈ B orequivalently n ( z − y ) + n ( y − x ) ∈ B which is the same as n ( z − x ) ∈ B ,precisely stating that x ≤ n z .In order to prove condition (C3) we need an extra assumption onthe subset B , namely that B n ⊆ B for every n ∈ N . This is clearly thecase for normed vector spaces and the previous considerations can besummarized as follows. Proposition 4.2.
Let ( X, + , be a group and suppose there are B ⊆ X and α : B → N satisfying for every x ∈ B , u ∈ X and n ∈ N thefollowing three conditions:(i) ∈ B ;(ii) if α ( x ) u ∈ B then u + x ∈ B (iii) if nu ∈ B then u ∈ B Then, the system ( X, ≤ n , ∂ n ) , defined as x ≤ n y whenever n ( y − x ) ∈ B and ∂ n ( x, y ) = α ( n ( y − x )) n , is a cartesian spacial fibrous preorderindexed by the natural numbers. Thus giving rise to an N -cartesianspace on the set X . In the following section we will analyse in more detail the exampleof a metric space. The example of a normed vector space, which has been generalized by the previous proposition, will be analysed furtherin Section 6. 5.
The structure of metric spaces
In this section we will see how to decompose the structure of a metricspace in terms of a cartesian spacial fibrous preorder. First, insteadof the non-negative real interval [0 , + ∞ [, we can take any preorder( E, ≤ ). Secondly, instead of the formula n − d ( x, y ) we will use maps p ( α, x, y ) = α − d ( x, y ) and g ( n ) = n so that n − d ( x, y ) is obtainedby taking p ( g ( n ) , x, y ). These maps take their values on the preorder( E, ≤ ) and are required to satisfy some conditions. The definition of x ≤ n y is recovered as g ( k ) ≤ p ( g ( n ) , x, y ) for some k ∈ I , which isthen used to define ∂ n ( x, y ) = k .Let ( E, ≤ ) be a preorder and B a set. We will say that a map p : E × B × B → E is a lax-left-associative Mal’tsev operation whenthe following three conditions are satisfied for all a, b ∈ E and x, y, z ∈ B : (1) a ≤ p ( a, x, x ),(2) p ( p ( a, x, y ) , y, z ) ≤ p ( a, x, z ),(3) if a ≤ b then p ( a, x, y ) ≤ p ( b, x, y ).Let ( I, · ,
1) be a unitary magma and ( E, ≤ ) a preorder. We will saythat a map g : I → E is a linking map if g ( n · ( k · m )) ≤ g ( n · m )for all n, m, k ∈ I . Proposition 5.1.
Let ( I, · , be a unitary magma and ( E, ≤ ) a pre-order. Every lax-left-associative Mal’tsev operation p : E × B × B → E together with a linking map g : I → E induces a topology τ on the set B determined by O ∈ τ ⇔ ∀ x ∈ O , ∃ n ∈ I, N ( n, x ) ⊆ O with N ( n, x ) = { y ∈ B | ∃ m ∈ I, g ( m ) ≤ p ( g ( n ) , x, y ) } .Proof. The proof makes use of Proposition 2.1 by showing that thesystem ( B, ≤ n , ∂ n ) is a cartesian spacial fibrous preorder with x ≤ n y ⇔ ∃ m ∈ I, g ( m ) ≤ p ( g ( n ) , x, y )and ∂ n ( x, y ) = m .Axiom (C1) holds because g ( n ) ≤ p ( g ( n ) , x, x ) for all n ∈ I and x ∈ B .In order to prove Axiom (C2) we consider x ≤ n y with ∂ n ( x, y ) = m such that g ( m ) ≤ p ( g ( n ) , x, y ) and y ≤ m z for which there exists m ′ = ∂ m ( y, z ) ∈ I such that g ( m ′ ) ≤ p ( g ( m ) , y, z ) . Under these assumptions, having in mind that p is a lax-left-associativeMal’tsev operation, we observe g ( m ′ ) ≤ p ( g ( m ) , y, z ) ≤ p ( p ( g ( n ) , x, y ) , y, z ) ≤ p ( g ( n ) , x, z )which shows that x ≤ n z .Axiom (C3) is a consequence of g being a linking map from which,in particular, we obtain g ( k · m ) ≤ g ( m )and g ( n · k ) ≤ g ( n ) . In the former case n = 1 while in the latter m = 1. This allows us toconclude that if x ≤ n · k y then x ≤ n y . Indeed, if there exists m ∈ I such that g ( m ) ≤ p ( g ( n · k ) , x, y ) then p ( g ( n · k ) , x, y ) ≤ p ( g ( n ) , x, y )and so g ( m ) ≤ p ( g ( n ) , x, y ), thus ensuring x ≤ n y . Similarly we provethat if x ≤ k · m y then x ≤ m y . (cid:3) We remark that the reflexivity of the preorder ( E, ≤ ) is never used,so that the previous result still holds for a set E equipped with atransitive relation. So, for example, the above result holds true if wereplace the preordered set ( E, ≤ ) by a semigroup ( E, ◦ ) and define thetransitive relation x < y as x ◦ y = x . As it is well known this relationfails to be reflexive when E is not an idempotent semigroup and failsto be anti-symmetric when E is not commutative. In addition, therequirement on the map g being a linking map could be replaced bythe condition that g ( n · m ) ≤ g ( n ) and g ( n · m ) ≤ g ( m ).The example of a metric space is obtained by letting E be the setof real numbers with the usual order, and taking the maps p ( a, x, y ) = a − d ( x, y ) and g ( n ) = n with I the unitary magma of natural numberswith usual multiplication.Another interesting example is obtained by taking E to be an orderedgroup ( E, + , , ≤ ) and I = ( I, · ,
1) a monoid. Suppose there exists amap g : I → E with g ( mkn ) ≤ g ( mn ) for all m, n, k ∈ I . Let B be anyset and consider a map δ : B × B → E such that δ ( x, x ) = 0 , for all x ∈ B (14) δ ( x, y ) + δ ( y, z ) ≥ δ ( x, z ) , for all x, y, z ∈ B . (15)This is a straightforward generalization of a metric space and we havethat p ( a, x, y ) = a − δ ( x, y ) is a lax-left-associative Mal’tsev operation. Let us now suppose that B is a group. Then, with g and E as before,for every map t : B → E such that t (0) = 0 and t ( u ) + t ( v ) ≥ t ( u + v ) , for all u, v ∈ B (16)we get a lax-left-associative Mal’tsev operation with p ( a, x, y ) = a − t ( y − x ). This is an immediate generalization for normed spaces. Inparticular, when t is a homomorphism, we get p ( a, x, y ) = a + t ( x ) − t ( y ).A simple procedure to construct unitary magmas which in generalare not associative is to start with an arbitrary non-empty set of in-dexes I , choose and element 1 ∈ I and consider a family of endo-maps µ n : I → I , indexed by the elements in I , such that µ n (1) = 1 and µ ( n ) = n for all n ∈ I . A binary operation is thus obtained as m · n = µ m ( n ).One final remark on notation. The notion of a linking map comesfrom the structure of a link which is part of work in progress. Thename lax-left-associative Mal’tsev operation is due to the fact thatwhen E = B and the preorder is the identity or discrete order, then, alax-left-associative Mal’tsev operation reduces to a = p ( a, x, x ) p ( p ( a, x, y ) , y, z ) = p ( a, x, z ) . If adding the respective two similar identities on the right p ( x, x, a ) = a and p ( z, x, a ) = p ( z, y, p ( y, x, a ), which make sense because E = B ,then we would obtain an associative Mal’tsev operation.6. Monoids and modules as cartesian spaces
An interesting special class of I -cartesian spaces is obtained by im-posing on the structuring maps γ and η of Theorem 3.1(e) the condition γ ( η ( n, x ) , y ) = γ ( η (1 , x ) , y ) · n for all y ∈ η ( n, x ) and for all n ∈ I and x ∈ X .We will analyse this condition by considering a monoid structureon the set B . This will allow us to decompose η ( n, x ) = x + η ( n, I -module structure on B thus providing a way toobtain η ( n,
0) as an n -scaling of η (1 , B is an I -module, all the information is encompassed in η (1 ,
0) and γ ( η (1 , , z ). Proposition 6.1.
Let ( I, · , be a unitary magma. Every monoid ( B, + , , equipped with a family of subsets S n ⊆ B together with maps α n : S n → I, n ∈ I such that(i) ∈ S n , for all n ∈ I (ii) for each n ∈ I , if a, a ′ ∈ S n and a ′ ∈ S α n ( a ) then a + a ′ ∈ S n (iii) S n · m ⊆ S n ∩ S m induces a topology τ on B determined as O ∈ τ ⇔ ∀ x ∈ O , ∃ n ∈ I, x + S n ⊆ O . Proof.
We make use of Proposition 2.1 by showing that the system( B, ( ≤ n ) n ∈ I , ( ∂ n ) n ∈ I ) is a cartesian spacial fibrous preorder with x ≤ n y ⇔ ∃ a ∈ S n , x + a = y and ∂ n ( x, y ) = α n ( a ) · n while noting that N ( n, x ) = { y ∈ B | x ≤ n y } is the same as x + S n = { x + a | a ∈ S n } .Axiom (C1) holds because 0 ∈ S n and hence x ≤ n x for all x ∈ B and n ∈ I .In proving axiom (C2) we observe that if x ≤ n y , that is, x + a = y for some a ∈ S n , and if y ≤ m z , with m = α n ( a ) · n , which means y + a ′ = z for some a ′ ∈ S m , then by condition ( iii ) we conclude that a ′ ∈ S n and a ′ ∈ S α n ( a ) . Condition ( ii ) now tells us that a + a ′ ∈ S n and hence x ≤ n z . Indeed, there exists a ′′ = a + a ′ ∈ S n such that x + a ′′ = z .Axiom (C3) is a straightforward consequence of condition ( iii ). If x ≤ n · m y , that is x + a = y for some a ∈ S n · m , then x ≤ n y and x ≤ m y since, by ( iii ), a ∈ S n and a ∈ S m . (cid:3) In particular, if there exists an action of I on B , in the sense of amap ξ : I × B → B such that for all n, m ∈ I and x, y ∈ B (1) ξ (1 , x ) = x , ξ ( n,
0) = 0,(2) ξ ( n, x + y ) = ξ ( n, x ) + ξ ( n, y )(3) ξ ( n · m, x ) = ξ ( n, ξ ( m, x )) = ξ ( m, ξ ( n, x )),then we may wonder if each S n in the previous proposition is deter-mined from S as S n = { ξ ( n, a ) | a ∈ S } .A monoid ( B, + ,
0) equipped with an action ξ of I on B in the senseabove is said to be an I -module and represented as ( B, + , , ξ ). Theorem 6.2.
Let ( I, · , be a unitary magma, ( B, + , , ξ ) an I -module and ( B, τ ) a topological space. The following are equivalent:(a) There exists S ⊆ B and α : S → I such that(i) ∈ S (ii) a + ξ ( α ( a ) , a ′ ) ∈ S , for all a, a ′ ∈ S for which τ is determined as O ∈ τ ⇔ ∀ x ∈ O , ∃ n ∈ I, x + S n ⊆ O with S n = { ξ ( n, a ) ∈ B | a ∈ S } .(b) There exists S ⊆ B such that(i) ∈ S (ii) S ⊆ { y ∈ B | ∃ n ∈ I, y + S n ⊆ S } , with S n = { ξ ( n, a ) ∈ B | a ∈ S } for which τ is determined as O ∈ τ ⇔ ∀ x ∈ O , ∃ n ∈ I, x + S n ⊆ O . (c) There exists S ∈ τ and γ : { ( U, x ) | x ∈ U ∈ τ } → I such that(i) ∈ S (ii) x + S n ∈ τ for all x ∈ B and n ∈ I with S n = { ξ ( n, a ) ∈ B | a ∈ S } (iii) if x ∈ U ∈ τ and n = γ ( U, x ) then x + S n ⊆ U .Moreover, under these conditions, ( B, τ ) is an I -cartesian space with x ≤ n y defined as y = x + ξ ( n, a ) for some a ∈ S and ∂ n ( x, y ) = α ( a ) · n if and only if ξ ( n, a ) ∈ S for all a ∈ S and n ∈ I .Proof. ( a ) ⇒ ( b ) We only need to prove that S is contained in { y ∈ B | ∃ n ∈ I, y + S n ⊆ S } with S n = { ξ ( n, a ) ∈ B | a ∈ S } . This is the same as proving that forevery a ∈ S there exists n ∈ I for which a + S n ⊆ S . It follows fromcondition ( a )( ii ) by choosing n = α ( a ). Indeed, for every a ∈ S , wehave a + S α ( a ) ⊆ S as a consequence of S α ( a ) = { ξ ( α ( a ) , a ′ ) | a ′ ∈ S } together with condition ( a )( ii ).( b ) ⇒ ( c ) Let x ∈ B and n ∈ I . In order to show that x + S n ∈ τ we consider z = x + ξ ( n, a ) for some a ∈ S and find m ∈ I for which z + S m ⊆ x + S n . Using condition ( b )( ii ) and given that a ∈ S , weobtain k ∈ I such that a + S k ⊆ S , or, in other words, a + ξ ( k, a ′ ) ∈ S for all a ′ ∈ S . We now take m = k · n and show z + S m ⊆ x + S n .Indeed,for every a ′ ∈ S , taking into account that ξ is an action, we observe z + ξ ( m, a ′ ) = z + ξ ( k · n, a ′ ) = z + ξ ( n, ξ ( k, a ′ ))= x + ξ ( n, a ) + ξ ( n, ξ ( k, a ′ ))= x + ξ ( n, a + ξ ( k, a ′ ))showing that z + ξ ( m, a ′ ) = x + ξ ( n, a + ξ ( k, a ′ )) is in x + S n because a + ξ ( k, a ′ ) ∈ S for all a ′ ∈ S . In particular, when x = 0 and n = 1 weobtain S ∈ τ . Thus, so far we have shown the existence of S satisfyingcondition ( c )( ii ). Condition ( c )( i ) is clear while Condition ( c )( iii ) isobtained by observing that if x ∈ U ∈ τ then, by the assumption on τ , there exists n ∈ I for which x + S n ⊆ U , and hence we choose γ ( U, x ) = n in order to satisfy condition ( c )( iii ).( c ) ⇒ ( a ) Given S ∈ τ , the map α : S → I is defined as α ( a ) = γ ( S, a ). From condition ( c )( iii ) it follows that a + S α ( a ) ⊆ S which means that for every a, a ′ ∈ S , a + ξ ( α ( a ) , a ′ ) ∈ S. This proves condition ( a )( ii ). Condition ( a )( i ) is clear. It remains toprove that τ is determined as prescribed. We observe, on the one hand, if O ∈ τ then for every x ∈ O there exists m = γ ( O , x ) ∈ I such that x + S m ⊆ O as asserted by condition ( c )( iii ). On the other hand, if O ⊆ B is such that for all x ∈ O there exists m ∈ I with x + S m ⊆ O ,then, because x + S m ∈ τ and O = [ x ∈O x + S m we conclude that O ∈ τ .Moreover, under these conditions ( B, τ ) is I -cartesian, where themap η (see item ( e ) in Theorem 3.1) is obtained as η ( n, x ) = x + S n ,if, and only if S n ⊂ S for all n ∈ I . Indeed, if S n ⊂ S for all n ∈ I weobserve:(i) x ∈ η ( n, x ), since 0 ∈ S and ξ ( n,
0) = 0.(ii) if x ∈ U ∈ τ and m = γ ( U, x ) then x + S m ⊆ U , which is thesame as η ( γ ( U, x ) , x ) ⊆ U .(iii) η ( n · m, x ) ⊆ η ( n, x ) ∩ η ( m, x ) follows from S n · m ⊆ S n ∩ S m which is a consequence of ξ ( n, a ) ∈ S for all n ∈ I and a ∈ S and the fact that ξ , being an action, is such that ξ ( n · m, a ) = ξ ( n, ξ ( m, a )) = ξ ( m, ξ ( n, a )).Conversely, if ( B, τ ) is I -cartesian with η ( n, x ) = x + S n then from η ( n · m, x ) ⊆ η ( n, x ) ∩ η ( m, x ) for all m, n ∈ I we deduce S n ⊆ S when m = 1. (cid:3) Examples
Some examples are presented so to illustrate the results on the pre-vious section.Theorem 6.2 can be specialized into the case when the unitary magmais the set of natural numbers with usual multiplication I = ( N , · , B, + ,
0) is any monoid considered as an I -module with ξ ( n, x ) = nx .In this case, any subset S ⊆ B together with a map α : S → N satisfy-ing the three conditions:(i) 0 ∈ S (ii) a + α ( a ) a ′ ∈ S for all a, a ′ ∈ S (iii) na ∈ S for all a ∈ S and n ∈ N .induces a topology τ on B generated by the system of open neighbour-hoods N ( n, x ) = { x + na | a ∈ S } .Note that condition ( iii ) is necessary so that the system ( B, ≤ n , ∂ n )with x ≤ n y whenever y = x + na for some a ∈ S and ∂ n ( x, y ) = α ( a ) n to be a cartesian spacial fibrous preorder. When B is a group it offersa further comparison with Proposition 4.2 which will be deepened in afuture work.When the map α : S → N is constant with α ( a ) = 1 then S must bea submonoid. Another example is obtained when B = ( B, + , · , ,
1) is a semi-ring.In this case we can take I = ( N , + ,
0) and choose any submonoid S for the additive structure, that is, S ⊆ B with 0 ∈ S and a + a ′ ∈ S forall a, a ′ ∈ S . Now, every choice of an element p ∈ S such that p n a ∈ S ,for all a ∈ S and n ∈ N , gives a topology on the set B generated bythe system of open neighbourhoods N ( n, x ) = { x + p n a | a ∈ S } . Indeed, we define an I -module with action ξ ( n, x ) = p n x , by takingsuccessive powers of p and considering p = 1. In this case the map α : S → N is the constant map α ( a ) = 0.When S is not closed under addition then we have to find for every a ∈ S a non-negative integer α ( a ) ∈ N such that a + p α ( a ) a ′ ∈ S for all a ′ ∈ S . The simple example of the real numbers ( R , + , · , , S = [0 ,
1[ and p = ,illustrates this situation.One more example with I = ( N , + ,
0) is constructed as follows.Take any monoid ( X, + ,
0) together with t : X → X , an endomor-phism of X , and consider a subset P ⊆ X with 0 ∈ P . Let B be the collection of all maps from X to X considered as a monoidwith component-wise addition, that is, B = ( X X , + , X ). With I =( N , + ,
0) we define an action on B as ξ ( n, f ) = t n ◦ f for every non-negative integer n and every map f : X → X . The action is defined forevery selected endomorphism t of X .In order to produce a topology on the set B we need to find a subset S ⊆ B together with a map α : S → N as in Theorem 6.2. We take S = { f ∈ B | ∃ n ∈ N , ∀ x ∈ X, ∀ y ∈ P, f ( x ) + t n ( y ) ∈ P } and put α ( f ) = n where n is the smallest non-negative integer suchthat for all x ∈ X and y ∈ P , f ( x ) + t n ( y ) ∈ P .Clearly, the constant null function 0 X : X → X is in S and moreover α (0 X ) = 0.The reason why f + t α ( f ) ◦ g ∈ S for all f, g ∈ S is mainly because t isan endomorphism and so t n ◦ g + t n + m = t n ◦ g + t n ◦ t m = t n ◦ ( g + t m ).Indeed, if f, g ∈ S with α ( f ) = n and α ( g ) = m then there exists k = n + m such that ( f + t n ◦ g )( x ) + t k ( y ) ∈ P for all x ∈ X and y ∈ P .The imposition that t n ◦ f ∈ S for all n ∈ N and f ∈ S is guaranteedas soon as the endomorphism t : X → X satisfies the condition t ( y ) ∈ P for all y ∈ P . In that case α ( t n ◦ f ) = n + α ( f ).A concrete simple example is the following. Take X = (]0 , , · ,
1) tobe the unit interval of real numbers with usual multiplication and notcontaining the element 0. Consider the endomorphism t ( x ) = √ x = x and let P =] , f ∈ S if and only if f ( x ) ∈ P forall x ∈ X . For every f ∈ S , α ( f ) = n with n being the smallest non-negative integer for which u · (cid:18) (cid:19) n > u = inf { f ( x ) | x ∈ X } . Note that u is always greater than , andlim n → + ∞ (cid:18) (cid:19) n = 1 . As a consequence of Theorem 6.2 we obtain a topology on B gener-ated from N ( n, u ) = { u + t n ◦ v | v ∈ S } as system of open neighbourhoods.8. Conclusion
The notion of cartesian spacial fibrous preorder has been introducedas a system ( X, ( ≤ i ) i ∈ I , ( ∂ i ) i ∈ I ) indexed over a unitary magma, I , andsatisfying conditions (C1), (C2) and (C3). Every such structure givesrise to a topological space (Proposition 2.1) and the spaces whichare thus obtained were called I -cartesian. It has been proven (The-orem 3.1) that a space ( X, τ ) is I -cartesian if and only if there exists amap η : I × X → τ such that x ∈ η ( i, x ) and η ( ij, x ) ⊆ η ( i, x ) ∩ η ( j, x )for all i, j ∈ I and x ∈ X , together with a map γ : { ( U, x ) | x ∈ U ∈ τ } → I such that η ( γ ( U, x ) , x ) ⊆ U . In a sequel to this work we will concen-trate our attention on the structure of morphisms between cartesianspacial fibrous preorders thus elevating the characterization of Theo-rem 3.1 to a categorical equivalence. The structure of metric spaceshas been decomposed as an indexing unitary magma, a preorder, alax-left-associative Mal’tsev operation, and a linking map (Proposition5.1). The special case of a normed vector space has been treated inthree different manners:(1) In Proposition 4.2 as a group with a distinguished subset con-taining the origin and satisfying some conditions (intuitively aconvex and bounded neighbourhood of the origin, namely theopen ball of radius one) together with a map intuitively mea-suring how close to the boundary an element is.(2) In Proposition 5.1 as a special case of a lax-left-associativeMal’tsev operation with p ( a, x, y ) = a − k y − x k .(3) In Theorem 6.2 as an I -module with action ξ ( n, x ) = n x , S = { a ∈ B | k a k < } and α ( a ) such that α ( a ) ≤ − k a k .A list of examples has been presented as a way to illustrate possible ap-plications. This is a first step for a systematic analysis on the structureof topological spaces. References [1] Aleksandrov, P., 1925. Zur Begr¨undung der n-dimensionalen mengentheoretis-chen Topologie. Mathematische Annalen 94, 296–308. (Cited on page 1.)[2] Aleksandrov, P., 1937 Diskrete R¨aume. Mat. Sbornik 2, 501–520. (Cited onpages 2 and 8.)[3] Aleksandrov, P., Hopf, H., 1935. Topologie. Springer-Verlag, Berlin. (Cited onpage 2.)[4] Bourbaki, N., 1951. El´ements de math´ematique II. Premi`ere partie. Lesstructures fondamentales de l’analyse. Livre III. Topologie g´en´erale. ChapitreI. Structures topologiques, second edition. Actualit´es scientifiques et indus-trielles, vol. 1142. Hermann, Paris. (Cited on page 2.)[5] Brown, R., 1988. Topology and Groupoids (or Topology, a Geometric Accountof General Topology, Homotopy Types and the Fundamental Groupoid), Pren-tice Hall, Europe. (Cited on page 2.)[6] Carath´eodory, C., 1918. Vorlesungen ¨uber reelle Funktionen. Teubner, Leipzig.(Cited on page 1.)[7] de la Vall´ee Poussin, C., 1916. Int´egrales de Lebesgue, fonctions d’ensemble,classes de Baire. Gauthier–Villars, Paris. (Cited on page 1.)[8] Dedekind, R., 1931. Gesammelte mathematische Werke, vol. 2 (R. Fricke, E.Noether, ¨O. Ore, Eds.). Vieweg, Braunschweig. (Cited on page 1.)[9] Fr´echet, M., 1906. Sur quelques points du Calcul fonctionnel. Rendiconti delCircolo Matematico di Palermo 22, 1–74. (Cited on page 1.)[10] Fr´echet, M., 1921. Sur les ensembles abstraits. Annales scientifiques de l’EcoleNormale Sup´erieure (3) 38, 341–388. (Cited on page 1.)[11] Hausdorff, F., 1914. Grundz¨uge der Mengenlehre. Veit, Leipzig. (Cited onpage 1.)[12] Hausdorff, F., 1927. Mengenlehre. de Gruyter, Berlin. (Cited on page 1.)[13] James, I.M. (Ed.), 1999. History of Topology. North-Holland, Amsterdam.(Cited on page 1.)[14] Kelley, J.L., 1955. General Topology. Van Nostrand, Princeton, NJ. (Cited onpage 2.)[15] Kuratowski, K., 1922. Sur l’op´eration ¯ A de l’analysis situs. Fundamenta Math-ematicae 3, 182–199. (Cited on page 1.)[16] Kuratowski, K., 1933. Topologie I. Espaces m´etrisables, espaces complets.Garasinski, Warsaw. (Cited on page 2.)[17] Lebesgue, H., 1902. Int´egrale, longueur, aire. Annali di matematica pura edapplicata (3) 7, 231–359. (Cited on page 1.)[18] Lefschetz, S., 1930. Topology. American Mathematical Society, New York.(Cited on page 2.)[19] Lefschetz, S., 1942. Algebraic Topology. American Mathematical Society, NewYork. (Cited on page 2.)[20] Martins-Ferreira, N., 2014. From A-spaces to arbitrary spaces via spatial fi-brous preorders. Math Texts 46 (Categorical methods in algebra and topology)221–235. (Cited on pages 2, 3, 4, and 8.)[21] D. Montgomery, L. Zippin, Topological transformation groups , Interscience(1955) (Cited on page 2.)[22] Moore, G.H., 2008. The emergence of open sets, closed sets, and limit points inanalysis and topology. Historia Mathematica 35, 220–241. (Cited on page 1.)[23] Riesz, F., 1905. Sur un th´eor`eme de M. Borel. Comptes rendus hebdomadairesdes s´eances de l’Acad´emie des Sciences, Paris 140, 224–226. (Cited on page 1.)[24] Sierpinski, W., 1934. Introduction to General Topology. University of Toronto,Toronto. (Cited on page 2.) [25] Tietze, H., 1923. Beitr¨age zur allgemeinen Topologie. I. Axiome f¨ur ver-schiedene Fassungen des Umgebungsbegriffs. Mathematische Annalen 88,290–312. (Cited on page 1.) School of Technology and Management, Centre for Rapid and Sus-tainable Product Development - CDRSP, Polytechnic Institute ofLeiria, P-2411-901 Leiria, Portugal.
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