aa r X i v : . [ m a t h . GN ] F e b GENERALIZED CELL STRUCTURES
ANA G. HERN ´ANDEZ-D ´AVILA, BENJAM´IN A. ITZ ´A-ORTIZ, AND ROC´IO LEONEL-G ´OMEZ
Abstract.
Cell structures were introduced by W. Debski and E. Tymchatyn as away to study some classes of topological spaces and their continuous functionsby means of discrete approximations. In this work we weaken the notion of cellstructure and prove that the resulting class of topological space admitting sucha generalized cell structure includes non-regular spaces.
Introduction
Cell structures may be thought of as devices to represent some topologicalspaces by means of discrete approximations, more precisely, to describe a space X as homeomorphic to a perfect image of an inverse limit of graphs. They wereintroduced in [1] for complete metric spaces and extended in [2] to topologicallycomplete spaces. The initial step in identifying a cell structure consists in consid-ering an inverse sequence of graphs, each of which have attached a reflexive andsymmetric relation. The vertices of each graph are the elements of a set G andthe edges are described by reflexive and symmetric subsets of G × G , or equiva-lently, the set of edges of G is an entourage of the diagonal of G . Such an inversesequence is said to be a cell structure when it admits a couple of properties: onethat allows the induced natural relation on its inverse limit to be an equivalencerelation (where two threads in the inverse limit are declared to be related if everypair of their corresponding components are related), and a second property thatallows the quotient of the inverse limit by its natural relation to be a perfect map-ping. This resulting quotient space is the said to admit, up to homeomorphism,a cell structure.In this paper we explore the natural question on the class of topological spaceobtained by admitting a weaker version of a cell structures, which we will callgeneralized cell structures, or g-cell structure for short. We are able to showthat there are topological spaces admitting such a g-cell structure but not a cellstructure. Furthermore, an explicit example of a nonregular space admitting ag-cell structure will be provided.Representing the structure of spaces as approximations of simpler more un-derstandable structures, such as graphs, is a fruitful idea used not only in topol-ogy [5], but in a variety of other areas such as spin networks [6], operator alge-bras [3], networks [7], among others. We divide this work in two sections. In Section 1 we introduce the notion ofgeneralized cell structure and prove the basic results needed in the rest of thepaper. The main results are given in Section 2.The authors gratefully acknowledge that this paper have benefited from stim-ulating conversations with Carlos Islas and Juan Manuel Burgos. The first authorreceived support for this work from CONACyT scholarship Num. 926215.1.
Preliminaries
In this section we give the definition of spaces admitting a g-cell structuretogether with some preliminary results which will be useful in the next section.For any set G , we will denote the diagonal of G as ∆ G = { ( x, x ) : x ∈ G } . When noconfusion arises, we will write ∆ rather than ∆ G . The set of natural numbers is asusual N = { , , , . . . } . Definition 1.1.
We say that an order pair (
G, r ) is a cellular graph if G is a non-empty topological space and r ⊂ G × G is a reflexive and symmetric relation on G . The vertices of the graph, the elements of G , will also be known as cells of thegraph, while the elements of r are the edges of the graph. Definition 1.2.
If (
G, r ) is a cellular graph and u ∈ G is a cell, we define theneighborhood of u to be the set B ( u, r ) = { v ∈ G : ( u, v ) ∈ r } . Therefore, the set B ( u, r ) is the set of vertices of G that are adjacent to u in G .More generally, for A ⊂ G we denote B ( A, r ) = [ a ∈ A B ( a, r ) . Following the terminology of [4, Section 8.1, pg. 426], given a cellular graph(
G, r ), since the relation r is reflexive, then the diagonal ∆ of G is a subset of r . Onthe other hand, since r is symmetric it follows that r is equal to its own inverserelation − r , where − r = { ( x, y ) : ( y, x ) ∈ r } . Conversely, given a entourage of thediagonal r of G , that is to say, a relation on G which contains ∆ and satisfies r = − r ,it follows that r is a reflexive and symmetric relation. Thus, we may characterizea cellular graph as a pair ( G, r ) where G is a nonempty topological space and r isan entourage of the diagonal. Finally, the composition of r + r = 2 r of the relation r with itself is defined as 2 r = { ( x, z ) : ∃ y ∈ G, ( x, y ) , ( y, z ) ∈ r } . In other words a pairof cells ( x, z ) belongs to 2 r if there exits a path of length 2 in the cellular graph( G, r ) joining them. Notice that 2 r is still a reflexive and symmetric relation of G . Furthermore, it is always true that B ( x, r ) ⊂ B ( x, r ) and the opposite inclusionfollows if and only if r is transitive. Definition 1.3.
Suppose that { ( G n , r n ) } is a sequence of cellular graphs and let { g n +1 n } n ∈ N be a family of continuous functions, called bonding maps, g n +1 n : G n +1 → G n such that ENERALIZED CELL STRUCTURES 3 • g nn is the identity on G n , • g ln = g mn ◦ g lm for n < m < l and • the bonding maps send edges to edges, that is, ( g n +1 n ( x ) , g n +1 n ( y )) ∈ r n when-ever ( x, y ) ∈ r n +1 .We will then say that n ( G n , r n ) , g n +1 n o is an inverse sequence of cellular graphs.The inverse limit of an inverse sequence of cellular graphs n ( G n , r n ) , g n +1 n o willbe denoted by G ∞ or lim ←− { G i , g i +1 i } . It is a nonempty subspace of Q n ∈ N G n with theproduct topology. Elements in G ∞ are called threads and the mappings g i : G ∞ → G i denote the restriction of the standard projection maps p i : Q n ∈ N G n → G i . Inother words, G ∞ = (cid:26) ¯ x = ( x n ) n ∈ N : ∀ j ≥ i, g ji ( x j ) = x i (cid:27) ⊂ Y n ∈ N G n . The topology in G ∞ is actually characterized by a basis, as the following resultshows. It is actually a special case of [4, Propositon 2.5.5]. We include a proof forcompleteness. Proposition 1.4.
Let G ∞ be the inverse limit of an inverse sequence of cellulargraphs n ( G n , r n ) , g n +1 n o . Then the collection of the sets of the form n g − i ( A i ) : i ∈ N and A i ⊂ G i is an open in the base of G i o defines a basis for the the topology of G ∞ . Proof.
Let ¯ x ∈ G ∞ and let U ⊂ G ∞ be an open set such that ¯ x ∈ U . By definition ofthe subspace topology, there exists an open set V ⊂ Q n ∈ N G i such that U = V ∩ G ∞ .Thus there exists an open set V = p − j ( U ) ∩ · · · ∩ p − j n ( U n ) in the base of Q n ∈ N G i satisfying ¯ x ∈ V ⊂ V . Let j = max { j i } . Since the bounding maps are continuous,for every i ∈ { , ..., n } , (cid:16) g jj i (cid:17) − ( U i ) is an open set in G j . Hence, T(cid:16) g jj i (cid:17) − ( U i ) is alsoan open set and ¯ x ∈ T(cid:16) g jj i (cid:17) − ( U i ). Let V j be an open set in the base of G j such that x j ∈ V j ⊂ T(cid:16) g jj i (cid:17) − ( U i ). Now, since g − j (cid:16)(cid:16) g jj i (cid:17) − ( U i ) (cid:17) = g − j i ( U i ) = G ∞ ∩ p − j i ( U i ), weget that ¯ x ∈ g − j ( V j ) ⊂ G ∞ ∩ p − j i ( U i ) ⊂ G ∞ ∩ V = U . (cid:3) Definition 1.5.
Let n ( G n , r n ) , g n +1 n o be an inverse sequence of cellular graphs andlet G ∞ be its inverse limit. We define the natural relation r on G ∞ by r = { ( ¯ x, ¯ y ) ∈ G ∞ × G ∞ : ∀ n, ( x n , y n ) ∈ r n } . In turns out that the inverse limits of an inverse sequence of cellular graphs isactually a cellular graph with the natural relation r on G ∞ given in Definition 1.5,as established in the following proposition. ANA G. HERN ´ANDEZ-D ´AVILA, BENJAM´IN A. ITZ ´A-ORTIZ, AND ROC´IO LEONEL-G ´OMEZ
Proposition 1.6. If G ∞ is the inverse limit of an inverse sequence of cellulargraphs then it is a cellular graph with respect its natural relation given in Defi-nition 1.5. Proof.
Let n ( G n , r n ) , g n +1 n o be an inverse sequence of cellular graphs and let G ∞ be its inverse limit. By definition, G ∞ is a topological space. We only need tocheck that r is reflexive and symmetric. This follows immediately since each r n isreflexive and symmetric. (cid:3) We are now ready to define a generalized cell structure. It consists of just onecondition which is a necessary condition implied by the first original conditionfor cell structures originally defined in [1]. It enables the natural relation inducedon the inverse limit of an inverse sequence of cellular graphs to be transitive andtherefore an equivalent relation.
Definition 1.7.
We say that an inverse sequence of cellular graphs n ( G n , r n ) , g n +1 n o is a generalized cell structure, or a g-cell structure for short, if its natural relation r given in Definition 1.5 is an equivalence relation. Proposition 1.8.
Let n ( G n , r n ) , g n +1 n o be an inverse sequence of cellular graphs andlet G ∞ be its inverse limit. Each of the following statement implies the next.(1) For each i ∈ N , the relation r i is an equivalence relation on G i .(2) For each thread ¯ x ∈ G ∞ and for each i ∈ N there exists j ≥ i such that g ji (cid:16) B (cid:16) x j , r j (cid:17)(cid:17) ⊂ B ( x i , r i ).(3) The inverse sequence n ( G n , r n ) , g n +1 n o is a g-cell structure. Proof.
To prove (1) implies (2), let ¯ x ∈ G ∞ and let i ∈ N . Take j = i . Since r j istransitive, by hypothesis, then B ( x j , r j ) = B ( x j , r j ). Since g ii is the identity, thisproves that (2) holds.Now, we prove (2) implies (3). By Proposition 1.6, the natural relation is re-flexive and symmetric. To check that it is a g-cell structure, the transitivity of thenatural relation is the only property missing. The proof follows from the sameargument given in the proof of [1, Lemma 3.1], which we repeat here for com-pleteness sake. Suppose that ( ¯ x, ¯ y ) , ( ¯ y, ¯ z ) ∈ r and let i ∈ N . Then there is j ≥ i suchthat g ji (cid:16) B (cid:16) x j , r j (cid:17)(cid:17) ⊂ B ( x i , r i ). Since ( x j , y j ) , ( y j , z j ) ∈ r j then z j ∈ B ( x j , r j ). There-fore z i = g ji ( z j ) ∈ B ( x i , r i ), that is, ( x i , z i ) ∈ r i . Thus ( ¯ x, ¯ z ) ∈ r , as wanted. (cid:3) Condition (2) in previous Proposition 1.8 may be interpreted as follows: Foreach ¯ x ∈ G ∞ and each i ∈ N , there exits j ≥ i such that g ji collapses any path oflength two beginning at the vertex x j and through three di ff erent vertices to anedge beginning at x i .Given an inverse sequence of cellular graphs, to prove that it is a g-cell struc-ture will usually be verified by checking condition (2) in Proposition 1.8. ENERALIZED CELL STRUCTURES 5
Notice that Definition 1.7 allows the spaces G n to be not necessary discrete.It would be trivial to propose a g-cell structure of any topological space X byconsidering the inverse sequence G n = X , r n = ∆ and the bonding maps g n +1 n areall the identity maps. Therefore, to avoid such trivialities, it becomes importantto require either that every G n has the discrete topology or that not all r n areequal to ∆ . The role of the bonding maps, evidently, becomes predominant, asthe following examples show. Example 1.9.
For each i ∈ N , consider G i = [0 ,
1] with the standard topology anddefine r i = ∆ ∪ n ( x, − x ) , (1 − x, x ) : 0 < x < o ∪ n (0 , ) , ( , , ( , , (1 , ) o , while the bonding maps are given by g n +1 n ( x ) = ( x if 0 ≤ x < − x if ≤ x ≤ . Then n ( G n , r n ) , g n +1 n o is a g-cell structure while none of the G n ’s is a discrete spaces.Observe that Example 1.9 also exhibits that the converse of the implication(1) ⇒ (2) in Proposition 1.8 is false. The following example will show how amodification on the bonding maps may produce the loss of the cell structure ona sequence of cellular graphs. Example 1.10.
Consider the cellular graphs { ( G n , r n ) } given in Example 1.9. Nowredefine all the bonding maps g n +1 n as the identity maps. Then the resultinginverse sequence is no longer a cell structure. Indeed, the constant sequences¯0 = (0 , , , . . . ), ¯ = ( , , , . . . ) and ¯1 = (1 , , , . . . ) belong to G ∞ . Furthermore,( ¯0 , ¯ ) ∈ r and ( ¯ , ¯1) ∈ r . However, ( ¯0 , ¯1) < r . Thus r is not an equivalence relation. Definition 1.11.
Let n ( G n , r n ) , g n +1 n o be a g-cell structure. Let G ∞ = lim ←− { G n , g n +1 n } and let r be the induced relation on G ∞ as given in Definition 1.5. We denote thequotient space G ∗ as G ∗ = G ∞ /r. We will say that a topological space X admits a g-cell structure if there exists ag-cell structure such that X is homeomorphic to G ∗ .In Example 1.9 one may show that G ∞ is homeomorphic to [0 , ] with the usualtopology and that ( ¯0 , ¯1) is the only nontrivial relation in r . Thus G ∞ is homeo-morphic to S . This gives a g-cell structure for S alternative to cell structureobtained in [1].When every r j is an equivalence relation, one might think that the inverse limitof G ∗ j = G j /r j is equal to G ∗ . This is not true, as the following example shows. ANA G. HERN ´ANDEZ-D ´AVILA, BENJAM´IN A. ITZ ´A-ORTIZ, AND ROC´IO LEONEL-G ´OMEZ
Example 1.12.
Consider the discrete spaces G n = (0 , ] ∩ Q together with r n = G n × G n . Define now the bonding maps g n +1 n to be the inclusion maps. Then G ∗ isempty, while the inverse limit of G ∗ n is the one point space.The above Example 1.12 may be slightly modified by taking G n = N , r n = G n × G n and all bonding maps to be the identity. Then again G ∗ = N is di ff erent fromthe inverse limit of the one-point space G ∗ n . Proposition 1.13.
Let n ( G n , r n ) , g n +1 n o n ∈ N be a g-cell structure. The quotient map π : G ∞ → G ∗ is closed if and only if for every ¯ x ∈ G ∞ and every open neighbour-hood A ⊂ G ∞ of B ( ¯ x, r ) there exists U in the base of G ∞ such that ¯ x ∈ U and B ( U , r ) ⊂ A . Proof.
Let ¯ x ∈ G ∞ and let A ⊂ G ∞ be an open subset in G ∞ such that B ( ¯ x, r ) ⊂ A .Define O = { ¯ z ∈ G ∞ : π ( ¯ z ) ∩ A , ∅} = { ¯ z ∈ G ∞ : B ( ¯ z, r ) ⊂ A } . If π is a closed map, by [4, Proposition 2.4.9], the set O is an open set and since¯ x ∈ O , by Proposition 1.4, there exists U in the base of G ∞ such that ¯ x ∈ U ⊂ O .By definition of O , for each ¯ y ∈ U , B ( ¯ y, r ) ⊂ A . Thus, B ( U , r ) ⊂ A . For the converse,if ¯ z ∈ O then, by hypothesis, there exists a basis element U such that ¯ z ∈ U and B ( U , r ) ⊂ A . Let ¯ y ∈ U then B ( ¯ y, r ) ⊂ B ( U , r ) ⊂ A and thus ¯ y ∈ O . Consequently U ⊂ O and so O is an open set. By [4, Proposition 2.4.9], π is a closed map. (cid:3) In Propositon 1.13, if G n has the discrete topology for all n , we obtained acondition similar to [1, Proposition 3.8] for cell structures as the following resultshows. Corollary 1.14.
Let n ( G n , r n ) , g n +1 n o n ∈ N be a g-cell structure where every G n has thediscrete topology. If for every ¯ x ∈ G ∞ and every open neighbourhood A ⊂ G ∞ of B ( x, r ) there exists j ∈ N such that B ( g − j ( x j ) , r ) ⊂ A then the collection { G ∗ \ π ( G ∞ \ V ) : V is open in G ∞ } is a basis of open sets for the topology of G ∗ . Proof.
Let [ ¯ x ] ∈ G ∗ and let U ⊂ G ∗ be an open set containing [ ¯ x ]. Since π − ( U ) isan open set in G ∞ , then by Proposition 1.4, for each ¯ y ∈ B ( ¯ x, r ) there exists i ¯ y suchthat g − i ¯ y ( y i ¯ y ) ⊂ π − ( U ). Let V = S (cid:26) g − i ¯ y ( y i ¯ y ) : ¯ y ∈ B ( ¯ x, r ) and g − i ¯ y ( y i ¯ y ) ⊂ π − ( U ) (cid:27) , then V is an open set G ∞ such that B ( ¯ x, r ) ⊂ V ⊂ π − ( U ). By Proposition 1.13, π ( G ∞ \V )is a closed set and thus G ∗ \ π ( G ∞ \V ) is an open set. On the other hand we have G ∞ \V ⊂ G ∞ \ B ( ¯ x, r ). This mean that B ( ¯ x, r ) ∩ ( G ∞ \V ) = ∅ and so [ ¯ x ] < π ( G ∞ \V ).Thus [ ¯ x ] ∈ G ∗ \ π ( G ∞ \V ). Finally, since G ∞ \ π − ( U ) ⊂ G ∞ \V , then π ( G ∞ \ π − ( U )) ⊂ π ( G ∞ \V ), so that G ∗ \ π ( G ∞ \V ) ⊂ G ∗ \ π ( G ∞ \ π − ( U )). Now, since π ( π − ( G ∗ \ U )) = G ∗ \ U and G ∞ \ π − ( U ) = π − ( G ∗ \ U ) then π ( G ∞ \ π − ( U )) = G ∗ \ U . We concludethat G ∗ \ π ( G ∞ \V ) ⊂ U . (cid:3) ENERALIZED CELL STRUCTURES 7
Lemma 1.15.
Let n ( G i , r i ) , g i +1 i o i ∈ N be a g-cell structure where each G i satisfies thatfor each x ∈ G i and for each open set U containing B ( x, r i ), there exists an openset O such that x ∈ O and B ( O, r ) ⊂ U . Let i ∈ N and let ¯ x = ( x , x , x , ... ) ∈ G ∞ .Then A x i ,U = n [ ¯ z ] ∈ G ∗ : z i ∈ U and there exists j > i such that ( z j , w j ) < r j , ∀ ¯ w ∈ g − i ( B ( U , r i ) \ U ) o is an open set in G ∗ and π ( ¯ x ) ∈ A x i ,U . Proof.
We will show that the condition in A x i ,U is independent of the represen-tative of the class. Let ¯ z ∈ G ∞ such that z i ∈ U and j > i satisfying ( z j , y j ) < r j for all ¯ y ∈ g − i ( B ( U , r i ) \ U ). Let ¯ w ∈ G ∞ such that ( ¯ w, ¯ z ) ∈ r . Since z i ∈ U , thereexists an open set O such that z i ∈ O and B ( O, r ) ⊂ U . Hence, since w i ∈ B ( O, r ),then w i ∈ U . By the definition of g-cell structure, there exits k > j such that g kj ( B ( z k , r k )) ⊂ B ( z j , r j ) and suppose that there exists ¯ y ∈ g − i ( B ( U , r i )) \ U such that( y k , w k ) ∈ r k . Then y k ∈ B ( z k , r k ), which implies ( z j , y j ) ∈ r j . This is a contradictionon the choosing of j . Thus, for all ¯ y ∈ g − i ( B ( U , r i ) \ U ), ( y k , w k ) < r k , this mean that¯ w satisfies the condition in A x i ,U .Now, we will prove that A x i ,U is an open set. Let ¯ z ∈ π − ( A x i ,U ), then z i ∈ U and there exists j > i such that ( z j , y j ) < r j for all ¯ y ∈ g − i ( B ( U , r i ) \ U ). Since z i ∈ U , then there exists an open set O such that z i ∈ O and B ( O, r i ) ⊂ U . Then¯ z ∈ g − i ( O ). We will show that g − i ( O ) ⊂ A x i ,U . Let ¯ u ∈ g − i ( O ), then u i ∈ O ⊂ U . Suppose that there exists ¯ y ∈ g − i ( B ( U , r i ) \ U ) such that ( y j , u j ) ∈ r j . Then( y i , u i ) ∈ r i . Hence, y i ∈ B ( u i , r i ) ⊂ B ( O, r i ) ⊂ U . This is a contradiction because y i ∈ B ( U , r i ) \ U . Consequently, ( y j , u j ) < r j for all ¯ y ∈ g − i ( B ( U , r i ) \ U ). Then [ ¯ u ] ∈ A x i ,U which implies that ¯ u ∈ π − ( A x i ,U ). Hence, π − ( A x i ,U ) is an open set, so that A x i ,U is an open set in G ∗ . (cid:3) Results
In this section we present the main results of this work. Since we do not requirethat for each thread ¯ x and each i ∈ N there is j ≥ i such that g ji ( B ( x i , r i ) is finite (acondition that implies that the quotient map G ∞ G ∗ to be a perfect map), webegin by giving alternative condition on the g-cell structure which implies G ∗ tobe Hausdor ff or normal. Theorem 2.1.
Let n ( G i , r i ) , g i +1 i o i ∈ N be a g-cell structure where each G i is a Haus-dor ff space and satisfies that for each x ∈ G i and for each open set U containing B ( x, r i ), there exists an open set O such that x ∈ O and B ( O, r i ) ⊂ U . Then thequotient space G ∗ is a Hausdor ff space. Proof.
Let π ( ¯ x ), π ( ¯ y ) in G ∗ such that π ( ¯ x ) , π ( ¯ y ) and ¯ x, ¯ y ∈ G ∞ . Then ( ¯ x, ¯ y ) < r andthus there exists i ∈ N such that ( x i , y i ) < r i . Since that G i is a Hausdor ff spacethere exists disjoint open sets U and V such that x i ∈ U and y i ∈ V . Define the ANA G. HERN ´ANDEZ-D ´AVILA, BENJAM´IN A. ITZ ´A-ORTIZ, AND ROC´IO LEONEL-G ´OMEZ following sets: A x i ,U = n [ ¯ z ] ∈ G ∗ : z i ∈ U and ∃ j > i such that ( z j , w j ) < r j , ∀ ¯ w ∈ g − i ( B ( U , r i ) \ U ) o ,B y i ,V = n [ ¯ z ] ∈ G ∗ : z i ∈ V and ∃ j > i such that ( z j , w j ) < r j , ∀ ¯ w ∈ g − i ( B ( V , r i ) \ V ) o . By the Lemma 1.15, A x i ,U and B y i ,V are open sets in G ∗ containing π ( ¯ x ) and π ( ¯ y ),respectively. We will show that A x i ,U and B y i ,V are disjoint sets. Suppose thatthere exists [ ¯ z ] ∈ G ∗ such that [ ¯ z ] ∈ A x i ∩ B y i , then there exists ¯ w and ¯ u in [ ¯ z ]such that ¯ w satisfies the condition in A x i ,U and ¯ u satisfies the condition in B y i ,V .Since ( w i , u i ) ∈ r i , then w i ∈ B ( V , r i ). But, U ∩ V = ∅ then w i ∈ B ( V , r i ) \ V and so¯ w ∈ g − i ( B ( V , r i ) \ V . Hence, ( w j , u j ) < r j for some j . This is a contradiction because¯ u and ¯ w are in the same equivalence class. (cid:3) Corollary 2.2. If n ( G i , r i ) , g i +1 i o is a g-cell structure where each G i is a discretespace then G ∗ is a Hausdor ff space. Proof.
Since G i is a discrete space, then it is a Hausdor ff space. Also, if x ∈ G i and U is an open set containing B ( x, r i ), then O = { x } is an open set containing x and B ( O, r i ) ⊂ U . By the Theorem 2.1, G ∗ is a Hausdor ff space. (cid:3) Theorem 2.3.
Let X be a space admitting a g-cell structure. Then X is normalprovided its g-cell structure satisfies the following properties: (i) Every G n is ametric space. (ii) For every ¯ x ∈ G ∞ and for every open neighborhood A ⊂ G ∞ of B ( ¯ x, r ) there exists U in the base of G ∞ such that ¯ x ∈ U and B ( U , r ) ⊂ A . Proof.
By Proposition 1.13, the quotient map π : G ∞ → G ∗ is closed. Since every G n is metric then Q n ∈ N G n is also metric, [4, Theorem 4.2.2, pg. 259]. Therefore G ∞ is a normal space and applying [4, Theorem 1.5.20, pg. 46] we get that G ∗ isa normal space and thus X is a normal space. (cid:3) Theorem 2.4.
Let { X k } k ∈ N be a sequence of topological spaces each admitting acellular structure and without isolated points. Then the wedge sum X = W k ∈ N X k does not admit a cell structure but does admit a g-cell structure. Proof.
For each k ∈ N , let us denote p k ∈ X k the distinguished base point in X k andlet q : F k ∈ N X k → W k ∈ N X k the quotient map which identifies all the points p k intoa single point x . Since each X k is a metrizable space (in fact completely metri-zable by [1, Theorem 3.6]), let ρ k be a metric in X k , for each k ∈ N . Suppose thatthere exists a countable neighborhood base { u i } i ∈ N of the point x . Since q − ( u k ) isan open set in ⊔ k ∈ N X k , then q − ( u k ) ∩ X k is an open set in X k . Also, q − ( u k ) ∩ X k isa non-degenerate set because x ∈ q − ( u k ) ∩ X k and X k is a topological space with-out isolated points. Then there exists y k ∈ q − ( u k ) ∩ X k \{ p k } . Put d k = ρ k ( p k , y k )and denote B k = { y k ∈ X k : ρ k ( y k , p k ) < d k } the open ball with center p k and radius d k . By definition of the mapping q we have q − ( q ( ⊔ B k )) = ⊔ B k , then by definitionof quotient topology, we have that q ( ⊔ B k ) is an open set in X = W k ∈ N X k which ENERALIZED CELL STRUCTURES 9 contains x . We claim that q ( ⊔ B k ) does not contain any of the elements of thebase { u i } i ∈ N . Indeed, if u m ⊂ q ( ⊔ B k ), then q − ( u m ) ⊂ q − ( q ( ⊔ B k )) = ⊔ B k . Hence, y m ∈ q − ( u m ) ∩ X m ⊂ B m , a contradiction. Thus, x does no admit a countableneighborhood base. Then X is not a first countable space. We conclude that X is not a metrizable space. Thus X does not admit a cell structure by [1, Theo-rem 3.6].For each k ∈ N , let ( ∗ ) k = { ( G ki , r ki ) , ( g k ) i +1 i } be a cell structure for X k built asin [1, Theorem 4.3]. The cell structure ( ∗ ) k determines a inverse limit which willbe denoted by G k ∞ and G ∗ k will denote the space determined by the cell structure.The i th projection map restricted to G k ∞ will be denoted by ( g k ) i . The quotientmap of G k ∞ onto G ∗ k will be denoted by π k while ϕ k will denote the homeomor-phism of G ∗ k onto X k .For each k ∈ N , fix ¯ x k = ( x k , x k , . . . ) ∈ ( φ k ◦ π k ) − ( p k ). Furthermore, since ( ∗ ) k is acell structure for X k , we may choose inductively j ki > j ki − the least natural numbersuch ( g k ) j ki j ki − (cid:16) B (cid:16) x kj ki , r kj ki (cid:17)(cid:17) ⊂ B (cid:16) x kj ki − , r kj ki − (cid:17) , where j k = 1. We define the followingsets: • G i = S k ∈ N G kj ki • A i = S k ∈ N r kj ki • B i = S k,t B (cid:18) ( g k ) j ki ( ϕ k ◦ π k ) − ( p k ) , r kj ki (cid:19) × B (cid:18) ( g t ) j ti ( ϕ t ◦ π t ) − ( p t ) , r tj ti (cid:19) For each i ∈ N , we will consider on G i the relation R i = A i ∪ B i . Notice that each R i is a reflexive relation, because A i contains to the diagonal of G i × G i . Also, R i isa symmetric relation, since A i and B i are symmetric. Thus, for each i ∈ N , ( G i , R i )is a cellular graph. We define, for each i ∈ N , the bonding maps as follow: let f ii be the identity on G i and f i +1 i : G i +1 → G i defined by f i +1 i ( x ) = ( g k ) j ki +1 j ki ( x ) , x ∈ G kj ki +1 . We will prove that (cid:16) f i +1 i ( x ) , f i +1 i ( y ) (cid:17) ∈ R i whenever ( x, y ) ∈ R i +1 . • If ( x, y ) ∈ A i +1 , then ( x, y ) ∈ r kj ki +1 for some k ∈ N . Since ( g k ) j ki +1 j ki is a com-position of bonding maps, we get that (cid:16) f i +1 i ( x ) , f i +1 i ( y ) (cid:17) ∈ r kj ki . Therefore, (cid:16) f i +1 i ( x ) , f i +1 i ( y ) (cid:17) ∈ A i ⊂ R i . • If ( x, y ) ∈ B i +1 , there exist l, k ∈ N such that x ∈ B (cid:16) ( g k ) j ki +1 ( ϕ k ◦ π k ) − ( p k ) , r kj ki +1 (cid:17) and y ∈ B (cid:16) ( g l ) j li +1 ( ϕ l ◦ π l ) − ( p l ) , r lj li +1 (cid:17) . Thus, there is z = ( g k ) j ki +1 ( ¯ z ) such that¯ z = ( ϕ k ◦ π k ) − ( p k ) and ( x, z ) ∈ r kj ki +1 . This implies that f i +1 i ( x ) = ( g k ) j kt +1 j ki ( x ) and f i +1 i ( z ) = ( g k ) j kt +1 j ki ( z ), therefore (cid:16) f i +1 i ( x ) , f i +1 i ( z ) (cid:17) ∈ r kj ki . Also, since f i +1 i ( z ) =( g k ) j ki +1 j ki (cid:16) ( g k ) j ki +1 ( ¯ z ) (cid:17) = ( g k ) j ki ( ¯ z ), we have f i +1 i ( x ) ∈ B (cid:16) ( g k ) j ki ( ϕ k ◦ π k ) − ( p k ) , r kj ki (cid:17) .In the same way, we get that f i +1 i ( y ) ∈ B (cid:16) ( g l ) j li ( ϕ l ◦ π l ) − ( p l ) , r lj ki (cid:17) . Hence, weconclude that (cid:16) f i +1 i ( x ) , f i +1 i ( y ) (cid:17) ∈ B i ⊂ R i .Therefore n(cid:16) G i , R i (cid:17) , f i +1 i o is a inverse sequence of cellular graphs and we will de-note G ∞ its inverse limit. Now to show that the condition of g-cell structure issatisfied, let ¯ x ∈ G ∞ and let i ∈ N . Then, for some t ∈ N , ¯ x ∈ lim ←− { F ti , f i +1 i } := F t ∞ ,where we set, for each i ∈ N , F ti = G tj ti . Notice that G t ∞ = F t ∞ , because the set { j ts : s ∈ N } is cofinal in N . Then, there is ¯ x ′ = ( x ′ , x ′ , . . . , x ′ i , . . . ) ∈ G t ∞ which corre-sponds to ¯ x = (cid:18) x tj t , x tj t , . . . , x tj ti , . . . (cid:19) ∈ lim ←− { F ti , f i +1 i } . For i ′ = j ti ∈ N , let k > j ti +2 > i suchthat ( g t ) j tk i ′ ( B ( x ′ j tk , r tj tk )) ⊂ B ( x ′ i ′ , r ti ′ ). We will show that f ki ( B ( x k , R k )) ⊂ B ( x i , R i ). Let y ∈ B ( x k , R k ). Notice that 2 R k = 2 A k ∪ B k ∪ ( A k + B k ) ∪ ( B k + A k ). • If ( x k , y ) ∈ A k , then there exists z ∈ G k such that ( x k , z ) ∈ A k and ( z, y ) ∈ A k .Since x k ∈ G tj tk , then ( x k , z ) ∈ r tj tk and necessarily z ∈ G tj tk . Thus ( z, y ) ∈ r tj tk .This implies that y ∈ B (cid:16) x k , r tj tk (cid:17) = B (cid:16) x ′ j tk , r tj tk (cid:17) . Therefore f ki ( y ) = ( g t ) j tk i ′ ( y ) ∈ B ( x ′ i ′ , r ti ′ ) = B ( x ′ j ti , r tj ti ) = B ( x i , r tj ti ) ⊂ B ( x i , A i ). We get that ( f ki ( y ) , f ki ( x k )) ∈ A i ⊂ R i . • If ( x k , y ) ∈ B k , since B k is transitive, we have ( x k , y ) ∈ B k and so we obtainthe result. • If ( x k , y ) ∈ A k + B k , there exists z ∈ G k such that ( x k , z ) ∈ A k and ( z, y ) ∈ B k .Since ( x k , z ) ∈ A k , then ( x k , z ) ∈ r tj tk . Hence (cid:18) ( g t ) j tk j ti +1 ( x k ) , ( g t ) j tk j ti +1 ( z ) (cid:19) ∈ r tj ti +1 . Onthe other hand, we have that z ∈ G tj tk and y ∈ G sj sk for some s ∈ N . Since( z, y ) ∈ B k , then ( g t ) j tk j ti +2 ( z ) ∈ B (cid:18) ( g t ) j ti +2 ( ϕ t ◦ π t ) − ( p t ) , r tj ti +2 (cid:19) and ( g s ) j sk j si ( y ) ∈ B (cid:18) ( g s ) j si ( ϕ s ◦ π s ) − ( p s ) , r sj si (cid:19) . Now, (( g t ) j tk j ti +2 ( z ) , w ) ∈ r tj ti +2 for some w ∈ ( g t ) j ti +2 ( ϕ t ◦ π t ) − ( p t ) and therefore ( w, ( g t ) j ti +2 ( ¯ x t )) ∈ r tj ti +2 . This mean that ( g t ) j tk j ti +2 ( z ) ∈ B (( g s ) j ti +2 ( ¯ x t )) , r tj ti +2 ) and by the hypothesis, ( g t ) j tk j ti +1 ( z ) ∈ B (( g t ) j ti +1 ( ¯ x t )) , r tj ti +1 ).We obtain ( g t ) j tk j ti +1 ( x k ) ∈ B (cid:18) ( g t ) j ti +1 ( ¯ x t ) , r tj ti +1 (cid:19) . Also, g j ti +1 j ti (cid:18) B (cid:18) ( g t ) j ti +1 ( ¯ x t ) , r tj ti +1 (cid:19)(cid:19) ⊂ B (cid:16) ( g t ) j ti ( ¯ x t ) , r j ti (cid:17) . Then f ki ( x k ) = g j tk j ti ( x k ) ∈ B (cid:18) ( g t ) j ti ( ¯ x t ) , r tj ti (cid:19) . We conclude that ENERALIZED CELL STRUCTURES 11 f ki ( x k ) ∈ B (cid:18) ( g t ) j ti ( ¯ x t ) , r tj ti (cid:19) ⊂ B (cid:18) ( g t ) j ti ( ϕ t ◦ π t ) − ( p t ) , r tj ti (cid:19) . Since f ki ( y ) = ( g s ) j sk j si ( y ) ∈ B (cid:12)(cid:12)(cid:12)(cid:12) (( g s ) j si ( ¯ x s ) , r sj si (cid:19) ⊂ B (cid:18) ( g s ) j si ( ϕ s ◦ π s ) − ( p s ) , r sj si (cid:19) , we get that (cid:16) f ki ( x k ) , f ki ( y ) (cid:17) ∈ B i ⊂ R i . • If ( x k , y ) ∈ B k + A k , there exists z ∈ G k such that ( x k , z ) ∈ B k and ( z, y ) ∈ A k . Since ( z, y ) ∈ A k , there exists s ∈ N such that ( z, y ) ∈ r sj sk . Hence(( g s ) j sk j si +1 ( y ) , ( g s ) j sk j si +1 ( z )) ∈ r sj si +1 . In the other hands, we have that z ∈ G sj sk and x k ∈ G tj tk . Since ( x k , z ) ∈ B k , then ( g s ) j sk j si +2 ( z ) ∈ B (( g s ) j si +2 ( ϕ s ◦ π s ) − ( p s ) , r sj si +2 )and ( g t ) j tk j ti ( x k ) ∈ B (( g t ) j ti ( ϕ t ◦ π t ) − ( p t ) , r tj ti ). Thus (( g s ) j sk j si +2 ( z ) , w ) ∈ r sj si +2 where w ∈ ( g s ) j si +2 ( ϕ s ◦ π s ) − ( p s ). Hence, ( w, ( g s ) j si +2 ( ¯ x s )) ∈ r sj si +2 . This implies that( g s ) j sk j si +2 ( z ) ∈ B (( g s ) j si +2 ( ¯ x s )) , r sj si +2 ) and by the hypothesis, we get that ( g s ) j sk j si +1 ( z ) ∈ B (( g s ) j si +1 ( ¯ x s )) , r sj si +1 ). Notice that ( g s ) j sk j si +1 ( y ) ∈ B (( g s ) j si +1 ( ¯ x s ) , r sj si +1 ). Also, wehave g j si +1 j si ( B (( g s ) j si +1 ( ¯ x s ) , r sj si +1 )) ⊂ B (( g s ) j si ( ¯ x s ) , r sj si ). Therefore f ki ( y ) = g j sk j si ( y ) ∈ B (( g s ) j si ( ¯ x s ) , r sj si ). We conclude that f ki ( y ) ∈ B (( g s ) j si ( ¯ x s ) , r sj si ) ⊂ B (( g s ) j si ( ϕ s ◦ π s ) − ( p s ) , r sj si ). Since f ki ( x k ) = ( g t ) j tk j ti ( x k ) ∈ B (( g t ) j ti ( ¯ x t ) , r tj ti ) ⊂ B (( g t ) j ti ( ϕ t ◦ π t ) − ( p t ) , r tj ti ),we get that ( f ki ( x k ) , f ki ( y )) ∈ B i ⊂ R i .We now define the following sets: A = { ( ¯ x, ¯ y ) ∈ G ∞ × G ∞ : ( x i , y i ) ∈ A i , for each i ∈ N } and B = { ( ¯ x, ¯ y ) ∈ G ∞ × G ∞ : ( x i , y i ) ∈ B i , for each i ∈ N } . Notice that since A is the induced relation on G ∞ by the reflexive and symmetricrelation A i on G i , for each i , we have that A is an equivalence relation. So we willconsider the quotient space G ∞ /A (which is in fact homeomorphic to ⊔ X k ) andwe will introduce and equivalence relation on G ∞ /A as follows: B ′ = n(cid:16) [ ¯ x ] A , [ ¯ y ] A (cid:17) ∈ G ∞ /A × G ∞ /A : ( ¯ x, ¯ y ) ∈ B o ∪ ∆ . We will prove that the definition of B ′ does not depend on the class representa-tive. It will su ffi ce to show that given ¯ u ∈ [ ¯ x ] A such that for each i ∈ N we have x i ∈ B (cid:16) ( g k ) j ki ( ϕ k ◦ π k ) − ( p k ) , r kj ki (cid:17) for some k ∈ N , implies that for each i ∈ N wealso have u i ∈ B (cid:16) ( g k ) j ki ( ϕ k ◦ π k ) − ( p k ) , r kj ki (cid:17) for some k ∈ N . For this purpose, fix an arbitrary i ∈ N . Since x i ∈ G i , there exists k ∈ N such that x i ∈ G kj ki . Then( u i , x i ) ∈ r kj ki and ( u i +1 , x i +1 ) ∈ r j ki +1 . Also, by assumption, there exists ¯ w ∈ G ∞ suchthat π k ( ¯ w ) = ϕ − k ( p k ) and ( x i +2 , w j ki +2 ) ∈ r kj ki +2 . Thus x i +2 ∈ B (cid:16) x kj ki +2 , r kj ki +2 (cid:17) and this im-plies that (cid:16) x i +1 , x kj ki +1 (cid:17) ∈ r kj ki +1 . Hence, u i +1 ∈ B (cid:16) x kj ki +1 , r kj ki +1 (cid:17) and so, ( u i , x kj ki ) ∈ r kj ki . Weget that u i ∈ B (cid:16) ( g k ) j ki ( ϕ k ◦ π k ) − ( p k ) , r kj ki (cid:17) , as wanted. Thus B ′ is well defined. Noticethat B ′ is an equivalence relation. Consider the funtion ϕ : ( G ∞ /A ) /B ′ → G ∞ /R defined as ϕ (cid:16)h [ ¯ x ] A i B ′ (cid:17) = [ ¯ x ] R . We will show tha ϕ is well defined. G ∞ G ∞ /A ( G ∞ /A ) /B ′ G ∞ /R π A ∪ B π A π B ′ ϕ Figure 1.
The spaces G ∞ /R and ( G ∞ /A ) /B are homeomorphicLet [ ¯ y ] A ∈ h [ ¯ x ] A i B ′ . If (cid:16) [ ¯ x ] A , [ ¯ y ] A (cid:17) ∈ B ′ \ ∆ , then ( ¯ x, ¯ y ) ∈ B ⊂ R . If [ ¯ x ] A = [ ¯ y ] A , then( x i , y i ) ∈ A i and thus ( ¯ x, ¯ y ) ∈ A ⊂ R . It follows that the diagram in Figure 1 com-mutes. Hence ϕ is continuous. We now show that ϕ is injective. Let (cid:16) [ ¯ x ] A , [ ¯ y ] A (cid:17) < B ′ , then ( ¯ x, ¯ y ) < A , since otherwise [ ¯ x ] A = [ ¯ y ] A and so (cid:16) [ ¯ x ] A , [ ¯ y ] A (cid:17) ∈ ∆ ⊂ B ′ , a con-tradiction. But also ( ¯ x, ¯ y ) < B since otherwise ( ¯ x, ¯ y ) ∈ B implies by definition that (cid:16) [ ¯ x ] A , [ ¯ y ] A (cid:17) ∈ B ′ , a contradiction. We conclude that [ ¯ x ] R , [ ¯ y ] R and thus ϕ is in-jective. It is clear that ϕ is surjective and its inverse [ ¯ x ] R h [ ¯ x ] A i B ′ is also con-tinuous by an analogous argument using a commutative diagram. Thus ϕ is ahomeomorphism (cid:3) In fact the space X in Theorem 2.4 is a normal space. In the next result weprove that more is true: there is a non-regular space admitting a g-cell structure.First, we build the g-cell structure which will determine a non-regular space. Definition 2.5.
Consider the following sets with the discrete topology. G = { a } ∪ n b k : k ∈ N o , G = { a } ∪ n b k : k ∈ N o ∪ C ∪ n d k : k ∈ N o where C = n c , , c , o and let L = 1. In general, for each i > L i = L i − + i − C ik = n c i,k , c i,k o with 1 ≤ k ≤ L i and G i = n a i o ∪ n b ki : k ∈ N o ∪ L i [ k =1 C ik ∪ n d ki : k ∈ N o . ENERALIZED CELL STRUCTURES 13
Now, for each i ∈ N , r i = n(cid:16) a i , b ki (cid:17) , (cid:16) b ki , a i (cid:17) : k ≥ i o ∪ n(cid:16) b ki , b ni (cid:17) : k, n ≥ i o ∪ n ( a, b ) : a, b ∈ C ik , k = 1 , ..., L i o ∪ n(cid:16) d ( i − j + ki , b ki (cid:17) , (cid:16) b ki , d ( i − j + ki (cid:17) : k < i, j ∈ N ∪ { } o ,g ii : G i → G i is the identity function on G i and g i +1 i : G i +1 → G i is defined as g i +1 i ( a ) = a i if a = a i +1 (1) b ki if a = b ki +1 (2) a i if a = c i +1 , (3) b ii if a = c i +1 , (4) c ri,k − if a = c ri +1 ,k , r = 1 , , k = 2 , ..., L i + 1 (5) d k − ( L i +1) i if a = c i +1 ,k , k = L i + 2 , ..., L i +1 (6) b k − ( L i +1) i if a = c i +1 ,k , k = L i + 2 , ..., L i +1 (7) d ( i − j +1)+ ni if a = d ki +1 , k = i · j + n for 0 < n < i, j ∈ N ∪ { } (8) a i if a = d ki +1 , k = i · j, j ∈ N ∪ { } (9)First we will prove the following lemma. Lemma 2.6.
The sequence n(cid:16) G i , r i (cid:17) , g i +1 i o i ∈ N given in Definition 2.5 is an inversesequence of cellular graphs. Furthermore, given ¯ w = ( w , w , w , ... ) ∈ G ∞ suchthat w i = d ki for some k, i ∈ N , then there exists j ∈ N such that w j ∈ C j s , forsome s ∈ N . Proof.
By definition, r i is a reflexive and symmetric relation for each i ∈ N . Also,by definition of the bonding functions, we get that { ( G i , r i ) , g i +1 i } i ∈ N satisfies thefirst two conditions of inverse system. We will show that the third condition issatistied. Let a, b ∈ G i +1 such that ( a, b ) ∈ r i +1 . • If a = a i +1 and b = b ki +1 with k ≥ i , then g i +1 i ( a ) = a i and g i +1 i ( b ) = b ki . Hence,( g i +1 i ( a ) , g i +1 i ( b )) ∈ r i . • If a = b ki +1 and b = b ni +1 , then g i +1 i ( b ki +1 ) = b ki and g i +1 i ( b ni +1 ) = b ni . Therefore,( g i +1 i ( a ) , g i +1 i ( b )) ∈ r i . • If a, b ∈ C i +1 k we can suppose that a = c i +1 ,k and b = c i +1 ,k . If k = 1,then g i +1 i ( a ) = a i and g i +1 i ( b ) = b ii . Thus, ( g i +1 i ( a ) , g i +1 i ( b )) ∈ r i . Now, if k ∈ { , ..., L i + 1 } , we get that g i +1 i ( a ) = c i,k − and g i +1 i ( b ) = c i,k − . Hence,( g i +1 i ( a ) , g i +1 i ( b )) ∈ r i . By last, if k ∈ { L i + 2 , ..., L i +1 } , then g i +1 i ( a ) = d k − ( L i +1) i and g i +1 i ( b ) = b k − ( L i +1) i . Therefore ( g i +1 i ( a ) , g i +1 i ( b )) ∈ r i . • Suppose that b = b ki +1 , with k < i + 1 , a = d i · j + ki +1 with j ∈ N ∪ { } . If i · j + k = i · r for some r ∈ N , then g i +1 i ( a ) = a i and g i +1 i ( b ) = b ki . So, we get that ( g i +1 i ( a ) , g i +1 i ( b )) ∈ r i . Now, if i · j + k = i · r + n with 0 < n < i , then g i +1 i ( a ) = d ( i − j +1)+ ni and g i +1 i ( b ) = b ki . Since that n < i , we conclude that( g i +1 i ( a ) , g i +1 i ( b )) ∈ r i .Consequently, we get that { ( G i , r i ) , g i +1 i } is an inverse system of cellular graphs.Let k ≤ i −
1, notice that w i +1 = c i +1 ,k + L i +1 , then we can take j = i + 1 and s = k + L j − + 1 and we get that w j ∈ C jk + L j − +1 . Now, we suppose that k > i − k as k = ( i − j + 1) + n for some j ∈ N ∪ { } and 0 1. By definition of the function g i +1 i we get that w i +1 = d i · j + ni +1 . Now, weconsider the preimage of w i +1 under the function g i +2 i +1 . We have two cases again.If i · j + n ≤ i , we get that w i +2 = c i +2 ,i · j + n + L i +1 +1 . Hence, we can take j = i + 2 and s = i · j + n + L j − + 1. This implies that w j ∈ C j i · j + n + L j − +1 . If i · j + n > i , then w i +2 = d ( i +1)( j − ni +2 . In general, if w i + t is not in some C ik , then w i + t = d ( i + t − j − t +1)+ ni + t .Taking m = j + 1, we have w i + m = d ni + m . Since 0 < n < i − ≤ i + m − 1, we get that w i + m +1 = c ii + m +1 ,n + L i + m +1 . Now, we can take j = i + m + 1 and s = n + L j − + 1 andwe get that w j ∈ C j n + L j − +1 . Consequently, for each ¯ w ∈ G ∞ such that w i = d ki with k, i ∈ N , there exists j ∈ N such that w j ∈ C js , for some s ∈ N . (cid:3) We will also need the following. Lemma 2.7. The sequence n(cid:16) G i , r i (cid:17) , g i +1 i o i ∈ N given in Definition 2.5 is a g-cellstructure. Proof. By Lemma 2.6, the system given in Definition 2.5 is an inverse system. Itremains to prove that the condition for a g-cell structure is satisfied. Let ¯ w =( w , w , w , ... ) ∈ G ∞ and i ∈ N . If w i = a i or w i = b ki with k ≥ i , then B ( w i , r i ) = { a i } ∪ { b ki : k ≥ i } and B ( w i , r i ) = { a i } ∪ { b ki : k ≥ i } . Now, if w i = b ki , with k < i , weget that B ( w i , r i ) = { b ki } ∪ { d ( i − j + ki : j ∈ N ∪ { }} and B ( w i , r i ) = { b ki } ∪ { d ( i − j + ki : j ∈ N ∪ { }} . If w i ∈ C ik for some k ∈ N , then B ( w i , r i ) = C ik and B ( w i , r i ) = C ik .Hence, if w i ∈ { a i } ∪ { b ki : k ∈ N } ∪ (cid:16)S L i k =1 C ik (cid:17) , then B ( w i , r i ) = B ( w i , r i ). Finally, if¯ w ∈ G ∞ such that w i = d ki for some k, i ∈ N . Then, by the Lemma 2.6, there exists j ∈ N such that w j ∈ C j s for some s ∈ N . Therefore, we have that B ( w j , r j ) = B ( w j , r j ). We conclude that for each i ∈ N and ¯ w ∈ G ∞ , there exists j ∈ N suchthat g ji ( B ( w j , r j )) ⊂ B ( w i , r i ). Consequently, { ( G i , r i ) , g i +1 i } i ∈ N is a g-cell structure. (cid:3) We are now ready to prove our last result. Theorem 2.8. There are spaces admitting a g-cell structure which are not regular. ENERALIZED CELL STRUCTURES 15 Proof. Consider the inverse sequence of cell graphs n(cid:16) G i , r i (cid:17) , g i +1 i o i ∈ N given in De-finition 2.5. By Lemma 2.7 it is a g-cell structure. Let ¯ a = ( a , a , a , ... ), A = n ( b k , b k , b k , ... ) : k ∈ N o and B = π ( A ). We will show that B is a closed set in G ∗ andit does not contain π ( ¯ a ). Let b k = ( b k , b k , b k , ... ) ∈ A for some k ∈ N . By definitionof the relation r k +1 , the element a k +1 is related with b tk +1 if and only if t ≥ k + 1.Hence, ( b kk +1 , a k +1 ) < r k +1 this implies that ( b k , ¯ a ) < r . Since k was chosen arbitrar-ily, ¯ a is not related with some element of A . Therefore, π ( ¯ a ) < B .Let ¯ x = ( x , x , x , ... ) ∈ π − ( G ∗ \ B ). We shall show that π − ( G ∗ \ B ) ⊂ G ∞ \ A . No-tice that if ¯ b ∈ π − ( G ∗ \ B ), then π (¯ b ) ∈ G ∗ \ B and thus π (¯ b ) < B = π ( A ). It resultthat ¯ b < A and this implies that ¯ b ∈ G ∞ \ A . Therefore, ¯ x < A . Now, we will showthat there exists k ∈ N such that for every n ∈ N , x k , b nk . Suppose that for every k ∈ N there exists n ∈ N such that x k = b nk . In particular for k = 1, there exists n such that x = b n . Since that x = g ( x ) and for k = 2 there exists n such that x = b n , then x = b n . Applying induction, we suppose that x t = b n t for some t ∈ N . Since x t = g t +1 t ( x t +1 ) and for k = t + 1 there exists n t +1 such that x t +1 = b n t +1 t +1 ,we get that x t +1 = b n t +1 . Then x t = b n t for every t ∈ N . Hence, ¯ x ∈ A which is acontradiction. So, there exists k ∈ N such that for each n ∈ N , x k , b nk .We shall show that g − k ( x k ) is an open set and ¯ x ∈ g − k ( x k ) ⊂ π − ( G ∗ \ B ). Let¯ w = ( w , w , w , ... ) ∈ g − k ( x k ), then w k = x k . Now, if ¯ w = ¯ a , then ¯ w ∈ π − ( G ∗ \ B )because π ( ¯ a ) ∈ G ∗ \ B . Suppose that ¯ w , ¯ a , then w k = d nk or w k = c rk,m with n, m ∈ N and r ∈ { , } . First we will consider the case which ¯ w satisfies w k = d nk forsome n ∈ N . By the Lemma 2.6, there exists l > k such that w l = c rl,t for some t ∈ N and r ∈ { , } . Therefore, by definition of r l , ( w l , b sl ) = ( c rl,t , b sl ) < r l for every s ∈ N . Then ( ¯ w, b s ) < r for each b s ∈ A this implies that π ( ¯ w ) < B . Hence, ¯ w ∈ π − ( G ∗ \ B ). Now, we will consider ¯ w such that w k = c rk,m for some m ∈ N and r ∈ { , } . Then ( c rk,m , b sk ) < r l for every s ∈ N . Similarly to the first case, we get that¯ w ∈ π − ( G ∗ \ B ). We conclude that g − k ( x k ) is an open set such that ¯ x ∈ g − k ( x k ) ⊂ π − ( G ∗ \ B ). Consequently π − ( G ∗ \ B ) is open this implies that G ∗ \ B is open in G ∗ .Therefore, B is closed in G ∗ .To show that G ∗ is not regular, we will prove that there are not disjoint opensets U and V in G ∗ such that π ( ¯ a ) ∈ U and B ⊂ V . Let U , V open sets in G ∗ suchthat π ( ¯ a ) ∈ U and B ⊂ V . Then π − ( U ) and π − ( V ) are open sets in G ∞ whichsatisfy π − ( π ( ¯ a )) ⊂ π − ( U ) and π − ( B ) ⊂ π − ( V ). Also, π − ( π ( ¯ a )) = { ¯ a } . Since¯ a ∈ π − ( U ) and π − ( U ) is open, there exists a basic open g − j ( u j ) such that ¯ a ∈ g − j ( u j ) ⊂ π − ( U ). This implies that a j = u j and thus g − j ( a j ) ⊂ π − ( U ). By defini-tion of r j , we have that ( a j , b jj ) ∈ r j . Consider the thread b j = ( b j , b j , b j , ..., b jj − , b jj , ... ).Then b j ∈ A and thus, b j ∈ π − ( B ) ⊂ π − ( V ). Since π − ( V ) is open, for b j thereexists a basic open g − i ( b ji ) such that b j ∈ g − i ( b ji ) ⊂ π − ( V ). If ¯ z ∈ g − i +1 ( b ji +1 ), then z i +1 = b ji +1 this implies that z i = g i +1 i ( b ji +1 ) = b ji . Hence, ¯ z ∈ g − i ( b ji ) and g − i +1 ( b ji +1 ) ⊂ g − i ( b ji ). Without losing generality, we suppose that i > j . Remem-ber that L i +1 = L i + i , then L i + 2 ≤ j + L i + 1 ≤ L i +1 . Applying the row 7 of thebonding function g i +1 i , we get that g i +1 i ( c i +1 ,j +( L i +1) ) = b ji . Now, we consider thethread ¯ c = (cid:16) b j , b j , b j , ..., b jj , ..., b ji , c i +1 ,j +( L i +1) , c i +2 ,j +( L i +1)+1 , c i +3 ,j +( L i +1)+2 , ... (cid:17) . Then ¯ c ∈ g − i ( b ji ) and thus, ¯ c ∈ π − ( V ). Define the thread ¯ d as¯ d = (cid:16) a , ..., g i +1 i − ( c i +1 ,j +( L i +1) ) , g i +1 i ( c i +1 ,j +( L i +1) ) , c i +1 ,j +( L i +1) , c i +2 ,j +( L i +1)+1 , ... (cid:17) . By definition of r i + k with k ∈ N , we have that (cid:16) c i + k,j +( L i + k ) , c i + k,j +( L i + k ) (cid:17) ∈ r i + k . Then (cid:16) ¯ d, ¯ c (cid:17) ∈ r and therefore ¯ d ∈ π − ( V ). Since i > j , we get that L i + 1 ≤ j + L i + 1 ≤ L i + i = L i +1 and applying the row 6 of the bonding function g i +1 i , it result that g i +1 i (cid:16) c i +1 ,j +( L i +1) (cid:17) = d ji . If j = i − 1, by row 9 of the function g ii − , we have that g ii − (cid:16) d ji (cid:17) = a i − = a j . In other case ( j < i − g ii − , g ii − (cid:16) d ji (cid:17) = d i − ji − . Considering now the function g i − i − , we have two cases again. If j = i − 2, then g i − i − (cid:16) d i − ji − (cid:17) = a i − = a j and in the case j < i − g i − i − (cid:16) d i − ji − (cid:17) = d i − ji − .Since i > j > 1, there exists 0 < l < i such that j = i − l . If we continue with thisprocess until j = i − l , we will get that g i − l +1 i − l (cid:16) d i − l + ji − l +1 (cid:17) = a i − l = a j . Therefore g ij (cid:16) d ji (cid:17) = a j . Consequently, ¯ d ∈ g − j ( a j ) and thus ¯ d ∈ π − ( U ). So, we get that π − ( U ) ∩ π − ( V ) , ∅ . This mean that there exists ¯ z ∈ G ∞ such that π ( ¯ z ) ∈ U y π ( ¯ z ) ∈ V .Hence, π ( ¯ z ) ∈ U ∩ V and this implies that U ∩ V , ∅ . Therefore, if U and V areopen sets in G ∗ such that π ( ¯ a ) ∈ U and B ⊂ V , then U ∩ V , ∅ . We conclude that G ∗ is not regular. (cid:3) Evidently, there is a net version of Definition 1.7, as was observed in [2] for cellstructures. Using this generalized definition one may check that uniform spaces( X, U ) have a nontrivial g-cell structure { ( G u , u ) } u ∈U where G u = X and g uu = I X forall u, v ∈ U . It would be interesting to determine what is the class of topologicalspaces admitting a g-cell structure. References [1] W. Debski and E.D. Tymchantyn. Cell structures and completely metrizable spaces and theirmappings. Colloq. Math , 147:181–194, 2017.[2] W. Debski and E.D. Tymchantyn. Cell structures and topologically complete spaces. Topol. Appl. , 239:293–307, 2018.[3] G. Elliott. On the Classification of Inductive Limits of Sequences of Semisimple Finite-Dimensional Algebras. J. Algebra , 38:29–44, 1976.[4] R. Engelking. General Topology . Sigma Series in Pure Mathematics, 1989. ENERALIZED CELL STRUCTURES 17 [5] S. Lefschetz. On compact spaces. Ann. Math. , 32:521–538, 1931.[6] C. Nash. Topology and physics—a historical essay. In I.M. James, editor, History of Topology ,chapter 12, pages 359–415. North Holland, Amsterdam, 1999.[7] M. Newman. Networks . Oxford University Press, Oxford, 2018. Centro de Investigaci ´on en Matem ´aticas, Universidad Aut ´onoma del Estado de Hidalgo,Pachuca, Hidalgo, Mexico Email address : [email protected],[email protected] Facultad de Ciencias, Universidad Nacional Aut ´onoma de M´exico, Mexico City, Mexico Email address ::