A remark on locally direct product subsets in a topological Cartesian space
aa r X i v : . [ m a t h . GN ] S e p A remark on locally direct product sub-sets in a topological Cartesian space
Hiroki Yagisita (Kyoto Sangyo University)
Abstract:Let X and Y be topological spaces. Let C be a path-connected closedset of X × Y . Suppose that C is locally direct product, that is, for any( a, b ) ∈ X × Y , there exist an open set U of X , an open set V of Y , a subset I of U and a subset J of V such that ( a, b ) ∈ U × V and C ∩ ( U × V ) = I × J hold. Then, in this memo, we show that C is globally so, that is, there exista subset A of X and a subset B of Y such that C = A × B holds. The proof is elementary. Here, we note that one might be able to thinkof a (perhaps, open) similar problem for a fiber product of locally trivial fiberspaces, not just for a direct product of topological spaces.In Appendix, we mentioned a simple example of a C ([0 , R )-manifoldthat cannot be embedded in the direct product ( C ([0 , R )) n as a C ([0 , R )-submanifold. In addition, we introduce the concept of topological 2-space,which is locally the direct product of topological spaces and an analog ofhomotopy category for topological 2-space. Finally, we raise a question onthe existence of an R n -Morse function and the existence of an R n -immersionin a finite-dimensional R n -Euclidean space. Here, we note that the problemof defining the concept of an R n roof :[Step 1]Let ( x, y ) is a continuous mapping from [0 ,
1] to C . Then, in this step,we show that ( x (0) , y (1)) ∈ C and ( x (1) , y (0)) ∈ C hold.Let T := { t ∈ [0 , | ∀ ( u, v ) ∈ [0 , t ] × [0 , t ] : ( x ( u ) , y ( v )) ∈ C } . Then, 0 ∈ T holds. Because C is a closed set of X × Y , T is a closed set of[0 , t ∈ [0 , ∩ T , there exists t ∈ ( t ,
1] such that[0 , t ] ⊂ T holds. Let t ∈ [0 , ∩ T. Let D := { ( u, v ) ∈ [0 , × [0 , | ( x ( u ) , y ( v )) ∈ C } . Then, D is a closed set of [0 , × [0 ,
1] and ∪ t ∈ [0 , { ( t, t ) } ⊂ D and [0 , t ] × [0 , t ] ⊂ D hold. Because C is a locally direct product set of X × Y , D is a locally directproduct set of [0 , × [0 , a, b ) ∈ [0 , × [0 , U and V of [0 , I of U and a subset J of V such that( a, b ) ∈ U × V and D ∩ ( U × V ) = I × J hold. So, there exist open sets U and V of [0 , I of U and asubset J of V such that ( t , t ) ∈ U × V and D ∩ ( U × V ) = I × J hold. Then, from t ∈ [0 , t ∈ ( t ,
1] such that[ t , t ] × [ t , t ] ⊂ U × V holds. So, from ∪ t ∈ [ t ,t ] { ( t, t ) } ⊂ D , ∪ t ∈ [ t ,t ] { ( t, t ) } ⊂ I × J holds. Hence,[ t , t ] × [ t , t ] ⊂ I × J holds. So, 2 t , t ] × [ t , t ] ⊂ D holds. Well, there exist n ∈ N , a , a , · · · , a n ∈ [0 , t ], open sets U , V , U , V , · · · , U n , V n of [0 ,
1] and sets I , J , I , J , · · · , I n , J n such that[0 , t ] × { t } ⊂ ∪ k =1 , , ··· ,n ( U k × V k )holds and for any k ∈ { , , · · · , n } , I k ⊂ U k , J k ⊂ V k , ( a k , t ) ∈ U k × V k and D ∩ ( U k × V k ) = I k × J k hold. Then, from 0 ≤ t < t ≤
1, there exists t ∈ ( t , t ] such that[ t , t ] ⊂ ∩ k =1 , , ··· ,n V k holds. Let S := { u ∈ [0 , t ] | [ u, t ] × [ t , t ] ⊂ D } . Then, from 0 ≤ t < t ≤ t ≤ t ∈ S holds. Because D is a closedset of [0 , × [0 , S is a closed set of [0 , t ]. Now, we show that for any u ∈ (0 , t ] ∩ S , there exists u ∈ [0 , u ) such that [ u , t ] ⊂ S holds. Let u ∈ (0 , t ] ∩ S . Then, there exists k ∈ { , , · · · , n } such that ( u , t ) ∈ U k × V k holds. From u ∈ (0 , t ], there exists u ∈ [0 , u ) such that [ u , u ] ⊂ U k holds. Because t ∈ V k and [ u , u ] ×{ t } ⊂ D hold, [ u , u ] ×{ t } ⊂ I k × J k holds. So, [ u , u ] ⊂ I k holds. On the other hand, from u ∈ S and [ t , t ] ⊂ V k , { u } × [ t , t ] ⊂ D ∩ ( U k × V k ) = I k × J k holds. So,[ t , t ] ⊂ J k holds. Therefore, [ u , u ] × [ t , t ] ⊂ D holds. Hence, u ∈ S holds. So, forany u ∈ (0 , t ] ∩ S , there exists u ∈ [0 , u ) such that [ u , t ] ⊂ S holds.Hence, S is a nonempty closed open set of [0 , t ]. Because S = [0 , t ] holds,[0 , t ] × [ t , t ] ⊂ D holds. Similarly, there exists t ∈ ( t , t ] such that3 t , t ] × [0 , t ] ⊂ D holds. Therefore, as we set t := min { t , t } , t ∈ ( t ,
1] and [0 , t ] × [0 , t ] ⊂ D hold. So, for any t ∈ [0 , ∩ T , there exists t ∈ ( t ,
1] such that [0 , t ] ⊂ T holds.Therefore, T is a nonempty closed open set of [0 , T = [0 , x (0) , y (1)) ∈ C and ( x (1) , y (0)) ∈ C hold. —[Step 2]Let A := { x ∈ X | ∃ y ∈ Y : ( x, y ) ∈ C } and B := { y ∈ Y | ∃ x ∈ X : ( x, y ) ∈ C } . Then, C ⊂ A × B holds. Let ( x , y ) ∈ A × B . Then, we show that( x , y ) ∈ C holds. There exist y ∈ Y and x ∈ X such that ( x , y ) ∈ C and ( x , y ) ∈ C hold. Because C is path-connected, in virtue of Step 1,( x , y ) ∈ C holds. (cid:4) Comment : Does there exist a C -manifold N such that for any C -manifolds M and M , N can not be embedded in M × M as a C -submanifold ?Kasuya proposed a candidate for such a compact C -manifold N . Our resultmay be useful to prove that it is such. For the definition of a C n -manifold,see [1] or [2].In [1], we proposed a candidate of a noncompact C -manifold N such that(1) for any C -manifolds M and M , N can not be embedded in M × M asa C -submanifold but (2) there exists k ∈ N such that N can be embeddedin the k -dimensional Euclidean R -space ( R k ) as an R -submanifold. — Remark : For deformation of R n -structures and C n -structures, see Kodairaand Spencer, Ann. of Math. , 74 (1961), 52-100. —
Problem : A C -holomorphically convex domain of C n is an n -dimensionalStein manifold. So, it is a closed C -submanifold of C n +1 . Let C be a C -holomorphically convex domain. Then, is C the direct product of some Steinmanifolds ? For the definition of C n -holomorphic convexity, see [2]. —Acknowledgment: As in Comment, Professor Naohiko Kasuya proposed it.This work was supported by JSPS KAKENHI Grant Number JP16K05245.4 ppendix 0 In [3], we gave a simple example of a connected metrizable 1-dimensional C ([0 , R )-manifold that cannot be embedded in the direct product space( C ([0 , R )) n as a C ([0 , R )-submanifold. Appendix 1
Definition 1 (2-space) : W := ( W, S ) is said to be a 2-space and S is said to be the system oflocal Cartesian neighborhoods of W , if it satisfies the followings.(1) W is a topological space. S is a set.(2) Let ϕ ∈ S . Then, ϕ is a homeomorphism from an open set W ϕ of W to the direct product of topological spaces U ϕ and V ϕ .(3) Let ϕ , ϕ ∈ S and c ∈ W ϕ ∩ W ϕ . Then, there exist an open set M of U ϕ , an open set N of V ϕ , a map f from M to U ϕ and a map g from N to V ϕ such that c ∈ ϕ − ( M × N ) ⊂ W ϕ holds and for any w ∈ ϕ − ( M × N ), ϕ ( w ) = ( f ( π U ( ϕ ( w ))) , g ( π V ( ϕ ( w ))))holds. Here, π U is the projection from U ϕ × V ϕ to U ϕ and π V is theprojection from U ϕ × V ϕ to V ϕ .(4) W = ∪ ϕ ∈ S W ϕ holds. Example 2 (locally direct product subset) :Let X and Y be topological spaces. Then, a locally direct product subsetof X × Y is a 2-space. Example 3 (2-product) :Let ( W , S ) and ( W , S ) be 2-spaces. Then, for any ( ϕ , ϕ ) ∈ S × S ,the map( ϕ , ϕ ) : W ,ϕ × W ,ϕ −→ ( U ,ϕ × U ,ϕ ) × ( V ,ϕ × V ,ϕ )is a homeomorphism. So, the 2-product W × W := ( W × W , S × S )is a 2-space. —5 efinition 4 (2-map) :Let h be a continuous map from a 2-space ( W , S ) to a 2-space ( W , S ).Then, h is said to be a 2-map, if it satisfies the following.Let c ∈ W , ϕ ∈ S , c ∈ W ,ϕ , ϕ ∈ S and h ( c ) ∈ W ,ϕ . Then, thereexist an open set M of U ,ϕ , an open set N of V ,ϕ , a map f from M to U ,ϕ and a map g from N to V ,ϕ such that c ∈ ϕ − ( M × N ) ⊂ h − ( W ,ϕ )holds and for any w ∈ ϕ − ( M × N ), ϕ ( h ( w )) = ( f ( π U ( ϕ ( w ))) , g ( π V ( ϕ ( w ))))holds. Here, π U is the projection from U ,ϕ × V ,ϕ to U ,ϕ and π V is theprojection from U ,ϕ × V ,ϕ to V ,ϕ . Definition 5 (2-homotopy) :Let h and h be 2-maps from a 2-space W to a 2-space W . Then, h and h said to be 2-homotopic, if there exists a continuous map H from [0 , × W to W such that for any t ∈ [0 , w ∈ W H ( t, w ) ∈ W is a2-map and for any w ∈ W , H (0 , w ) = h ( w ) and H (1 , w ) = h ( w ) hold. Example 6 :2-spaces { } × ( R / Z ) and ( R / Z ) × { } are homeomorphic. However, theyare not 2-homotopy equivalent. — Problem :(1) For a compact Hausdorff space T , define T -space, T -product, T -mapand T -homotopy.(2) Do there exist a contractible compact Hausdorff space T and T -spacessuch that the T -spaces are homeomorphic but not T -homotopy equivalent ?— Appendix 2