A remark on locally direct product subsets in a topological Cartesian space

See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/336086130 A remark on locally direct product subsets in a topological Cartesian space Preprint · September 2019 DOI: 10.13140/RG.2.2.10845.97769 CITATIONS 0 2 authors, including: Hiroki Yagisita Kyoto Sangyo University 10 PUBLICATIONS22 CITATIONS SEE PROFILE All content following this page was uploaded by Hiroki Yagisita on 27 September 2019. The user has requested enhancement of the downloaded file. A remark on locally direct product subsets in a topological Cartesian space Hiroki Yagisita (Kyoto Sangyo University) Abstract: Let Xand Ybe topological spaces. Let Cbe a path-connected closed set of X×Y. Suppose that Cis locally direct product, that is, for any (a, b)∈X×Y, there exist an open set Uof X, an open set Vof Y, a subset Iof Uand a subset Jof Vsuch that (a, b)∈U×Vand C∩(U×V) = I×J hold. Then, in this memo, we show that Cis globally so, that is, there exist a subset Aof Xand a subset Bof Ysuch that C=A×B holds. The proof is elementary. Here, we note that one might be able to think of a (perhaps, open) similar problem for a fiber product of locally trivial fiber spaces, not just for a direct product of topological spaces. In Appendix, we mentioned a simple example of a C([0,1]; R)-manifold that cannot be embedded in the direct product (C([0,1]; R))nas a C([0,1]; R)- submanifold. In addition, we introduce the concept of topological 2-space, which is locally the direct product of topological spaces and an analog of homotopy category for topological 2-space. Finally, we raise a question on the existence of an Rn-Morse function and the existence of an Rn-immersion in a finite-dimensional Rn-Euclidean space. Here, we note that the problem of defining the concept of an Rn-handlebody may also be considered. Keywords: Morse-Bott index, handle decomposition, universal covering space, foliated manifold, fiber bundle, fibration, holomorphic foliation, Stein manifold. The related literature: “https://www.researchgate.net/profile/Hiroki Yagisita” 1 Proof : [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)) ∈Cand (x(1), y(0)) ∈Chold. Let T:= {t∈[0,1] | ∀ (u, v)∈[0, t]×[0, t] : (x(u), y(v)) ∈C}. Then, 0 ∈Tholds. Because Cis a closed set of X×Y,Tis a closed set of [0,1]. We show that for any t0∈[0,1) ∩T, there exists t1∈(t0,1] such that [0, t1]⊂Tholds. Let t0∈[0,1) ∩T. Let D:= {(u, v)∈[0,1] ×[0,1] |(x(u), y(v)) ∈C}. Then, Dis a closed set of [0,1] ×[0,1] and ∪t∈[0,1]{(t, t)} ⊂ D and [0, t0]×[0, t0]⊂D hold. Because Cis a locally direct product set of X×Y,Dis a locally direct product set of [0,1] ×[0,1]. That is, for any (a, b)∈[0,1] ×[0,1], there exist open sets Uand Vof [0,1], a subset Iof Uand a subset Jof Vsuch that (a, b)∈U×Vand D∩(U×V) = I×J hold. So, there exist open sets U0and V0of [0,1], a subset I0of U0and a subset J0of V0such that (t0, t0)∈U0×V0and D∩(U0×V0) = I0×J0 hold. Then, from t0∈[0,1), there exists t2∈(t0,1] such that [t0, t2]×[t0, t2]⊂U0×V0 holds. So, from ∪t∈[t0,t2]{(t, t)} ⊂ D,∪t∈[t0,t2]{(t, t)} ⊂ I0×J0holds. Hence, [t0, t2]×[t0, t2]⊂I0×J0holds. So, 2 [t0, t2]×[t0, t2]⊂D holds. Well, there exist n∈N,a1, a2,· · · , an∈[0, t0], open sets U1, V1, U2, V2,· · · , Un, Vn of [0,1] and sets I1, J1, I2, J2,· · · , In, Jnsuch that [0, t0]× {t0} ⊂ ∪k=1,2,··· ,n(Uk×Vk) holds and for any k∈ {1,2,· · · , n},Ik⊂Uk,Jk⊂Vk, (ak, t0)∈Uk×Vkand D∩(Uk×Vk) = Ik×Jk hold. Then, from 0 ≤t0< t2≤1, there exists t3∈(t0, t2] such that [t0, t3]⊂ ∩k=1,2,··· ,nVk holds. Let S:= {u∈[0, t0]|[u, t0]×[t0, t3]⊂D}. Then, from 0 ≤t0< t3≤t2≤1, t0∈Sholds. Because Dis a closed set of [0,1] ×[0,1], Sis a closed set of [0, t0]. Now, we show that for any u0∈(0, t0]∩S, there exists u1∈[0, u0) such that [u1, t0]⊂Sholds. Let u0∈ (0, t0]∩S. Then, there exists k0∈ {1,2,· · · , n}such that (u0, t0)∈Uk0×Vk0 holds. From u0∈(0, t0], there exists u1∈[0, u0) such that [u1, u0]⊂Uk0 holds. Because t0∈Vk0and [u1, u0]×{t0} ⊂ Dhold, [u1, u0]×{t0} ⊂ Ik0×Jk0 holds. So, [u1, u0]⊂Ik0 holds. On the other hand, from u0∈Sand [t0, t3]⊂Vk0,{u0} × [t0, t3]⊂ D∩(Uk0×Vk0) = Ik0×Jk0holds. So, [t0, t3]⊂Jk0 holds. Therefore, [u1, u0]×[t0, t3]⊂Dholds. Hence, u1∈Sholds. So, for any u0∈(0, t0]∩S, there exists u1∈[0, u0) such that [u1, t0]⊂Sholds. Hence, Sis a nonempty closed open set of [0, t0]. Because S= [0, t0] holds, [0, t0]×[t0, t3]⊂D holds. Similarly, there exists t4∈(t0, t2] such that 3 [t0, t4]×[0, t0]⊂D holds. Therefore, as we set t1:= min{t3, t4},t1∈(t0,1] and [0, t1]×[0, t1]⊂D hold. So, for any t0∈[0,1) ∩T, there exists t1∈(t0,1] such that [0, t1]⊂T holds. Therefore, Tis a nonempty closed open set of [0,1]. Because T= [0,1] holds, (x(0), y(1)) ∈Cand (x(1), y(0)) ∈Chold. — [Step 2] Let A:= {x∈X| ∃ y∈Y: (x, y)∈C} and B:= {y∈Y| ∃ x∈X: (x, y)∈C}. Then, C⊂A×Bholds. Let (x0, y1)∈A×B. Then, we show that (x0, y1)∈Cholds. There exist y0∈Yand x1∈Xsuch that (x0, y0)∈C and (x1, y1)∈Chold. Because Cis path-connected, in virtue of Step 1, (x0, y1)∈Cholds. ■ Comment : Does there exist a C2-manifold Nsuch that for any C-manifolds M1and M2,Ncan not be embedded in M1×M2as a C2-submanifold ? Kasuya proposed a candidate for such a compact C2-manifold N. Our result may be useful to prove that it is such. For the definition of a Cn-manifold, see [1] or [2]. In [1], we proposed a candidate of a noncompact C2-manifold Nsuch that (1) for any C-manifolds M1and M2,Ncan not be embedded in M1×M2as aC2-submanifold but (2) there exists k∈Nsuch that Ncan be embedded in the k-dimensional Euclidean R2-space (Rk)2as an R2-submanifold. — Remark : For deformation of Rn-structures and Cn-structures, see Kodaira and Spencer, Ann. of Math., 74 (1961), 52-100. — Problem : A C-holomorphically convex domain of Cnis an n-dimensional Stein manifold. So, it is a closed C-submanifold of C2n+1. Let Cbe a C2holomorphically convex domain. Then, is Cthe direct product of some Stein manifolds ? For the definition of Cn-holomorphic convexity, see [2]. — Acknowledgment: As in Comment, Professor Naohiko Kasuya proposed it. This work was supported by JSPS KAKENHI Grant Number JP16K05245. 4 Appendix 0 In [3], we gave a simple example of a connected metrizable 1-dimensional C([0,1]; R)-manifold that cannot be embedded in the direct product space (C([0,1]; R))nas a C([0,1]; R)-submanifold. Appendix 1 Definition 1 (2-space) : W:= (W, S) is said to be a 2-space and Sis said to be the system of local Cartesian neighborhoods of W, if it satisfies the followings. (1) Wis a topological space. Sis a set. (2) Let φ∈S. Then, φis a homeomorphism from an open set Wφof W to the direct product of topological spaces Uφand Vφ. (3) Let φ1, φ2∈Sand c∈Wφ1∩Wφ2. Then, there exist an open set M1of Uφ1, an open set N1of Vφ1, a map ffrom M1to Uφ2and a map gfrom N1to Vφ2such that c∈φ−1 1(M1×N1)⊂Wφ2holds and for any w∈φ−1 1(M1×N1), φ2(w) = (f(πU(φ1(w))), g(πV(φ1(w)))) holds. Here, πUis the projection from Uφ1×Vφ1to Uφ1and πVis the projection from Uφ1×Vφ1to Vφ1. (4) W=∪φ∈SWφholds. Example 2 (locally direct product subset) : Let Xand Ybe topological spaces. Then, a locally direct product subset of X×Yis a 2-space. Example 3 (2-product) : Let (W1, S1) and (W2, S2) be 2-spaces. Then, for any (φ1, φ2)∈S1×S2, the map (φ1, φ2) : W1,φ1×W2,φ2−→ (U1,φ1×U2,φ2)×(V1,φ1×V2,φ2) is a homeomorphism. So, the 2-product W1×2W2:= (W1×W2, S1×S2) is a 2-space. — 5 Definition 4 (2-map) : Let hbe a continuous map from a 2-space (W1, S1) to a 2-space (W2, S2). Then, his said to be a 2-map, if it satisfies the following. Let c∈W1,φ1∈S1,c∈W1,φ1,φ2∈S2and h(c)∈W2,φ2. Then, there exist an open set M1of U1,φ1, an open set N1of V1,φ1, a map ffrom M1to U2,φ2and a map gfrom N1to V2,φ2such that c∈φ−1 1(M1×N1)⊂h−1(W2,φ2) holds and for any w∈φ−1 1(M1×N1), φ2(h(w)) = (f(πU(φ1(w))), g(πV(φ1(w)))) holds. Here, πUis the projection from U1,φ1×V1,φ1to U1,φ1and πVis the projection from U1,φ1×V1,φ1to V1,φ1. Definition 5 (2-homotopy) : Let h0and h1be 2-maps from a 2-space W1to a 2-space W2. Then, h0and h1said to be 2-homotopic, if there exists a continuous map Hfrom [0,1]×W1 to W2such that for any t∈[0,1], the map w1∈W17→ H(t, w1)∈W2is a 2-map and for any w1∈W1,H(0, w1) = h0(w1) and H(1, w1) = h1(w1) hold. Example 6 : 2-spaces {0}×(R/Z) and (R/Z)×{0}are homeomorphic. However, they are not 2-homotopy equivalent. — Problem : (1) For a compact Hausdorff space T, define T-space, T-product, T-map and T-homotopy. (2) Do there exist a contractible compact Hausdorff space Tand T-spaces such that the T-spaces are homeomorphic but not T-homotopy equivalent ? — Appendix 2 Let Mbe a paracompact connected Rn-manifold. Let (f1, f2,· · · , fn) be an Rn-map from Mto Rn. Suppose that for any local Cartesian neighborhood U1×U2×· · ·×Un, (a1, a2,· · · , an)∈U1×U2×· · ·×Unand k∈ {1,2,· · · , n}, the map xk∈Uk7→ fk(a1, a2,· · · , ak−1, xk, ak+1, ak+2,· · · , an)∈R is a Morse function on Uk. Then, does there exist m∈Nsuch that Mcan be Rn-immersed in the m-dimensional Rn-affine space (Rm)n? Related problem (Probably easy) : Let Mbe a compact foliation. Then, does there exist a Bott-Morse function fsuch that a critical manifold of fis a leaf of Mand a leaf of Mis contained in a level set of f? — 6 References [1] H. Yagisita, Holomorphic differential forms of complex manifolds on commutative Banach algebras and a few related problems, preprint. [2] H. Yagisita, Cartan-Thullen theorem for a Cn-holomorphic convexity and a related problem, preprint. [3] H. Yagisita, A manifold on the real commutative Banach algebra C([0,1]; R) that cannot be embedded in the finite-dimensional Euclidean space C([0,1]; Rn), preprint. [4] H. Yagisita, Finite-dimensional complex manifolds on commutative Banach algebras and continuous families of compact complex manifolds, Complex Manifolds, 6 (2019), 228-264. The related literature: “https://www.researchgate.net/profile/Hiroki Yagisita” 7 View publication statsView publication stats