Special embeddings of finite-dimensional compacta in Euclidean spaces

📝 Abstract

If $g$ is a map from a space $X$ into $\mathbb R^m$ and $z\not\in g(X) $, let $P_{2,1,m}(g,z)$ be the set of all lines $\Pi^1\subset\mathbb R^m$ containing $z$ such that $|g^{-1}(\Pi^1)|\geq 2 $. We prove that for any $n $-dimensional metric compactum $X$ the functions $g\colon X\to\mathbb R^m $, where $m\geq 2n+1 $, with $\dim P_{2,1,m}(g,z)\leq 0$ for all $z\not\in g(X)$ form a dense $G_\delta $-subset of the function space $C(X,\mathbb R^m) $. A parametric version of the above theorem is also provided.

💡 Analysis

If $g$ is a map from a space $X$ into $\mathbb R^m$ and $z\not\in g(X) $, let $P_{2,1,m}(g,z)$ be the set of all lines $\Pi^1\subset\mathbb R^m$ containing $z$ such that $|g^{-1}(\Pi^1)|\geq 2 $. We prove that for any $n $-dimensional metric compactum $X$ the functions $g\colon X\to\mathbb R^m $, where $m\geq 2n+1 $, with $\dim P_{2,1,m}(g,z)\leq 0$ for all $z\not\in g(X)$ form a dense $G_\delta $-subset of the function space $C(X,\mathbb R^m) $. A parametric version of the above theorem is also provided.

📄 Content

arXiv:1010.4838v1 [math.GN] 23 Oct 2010 SPECIAL EMBEDDINGS OF FINITE-DIMENSIONAL COMPACTA IN EUCLIDEAN SPACES SEMEON BOGATYI AND VESKO VALOV Abstract. If g is a map from a space X into Rm and z ̸∈g(X), let P2,1,m(g, z) be the set of all lines Π1 ⊂Rm containing z such that |g−1(Π1)| ≥2. We prove that for any n-dimensional metric compactum X the functions g : X →Rm, where m ≥2n + 1, with dim P2,1,m(g, z) ≤0 for all z ̸∈g(X) form a dense Gδ-subset of the function space C(X, Rm). A parametric version of the above theorem is also provided.

1. Introduction In this paper we assume that all topological spaces are metrizable and all single-valued maps are continuous. Everywhere below by Mm,d we denote the space of all d-dimensional planes Πd (br., d-planes) in Rm. If g is a map from a space X into Rm, q is an integer and z ̸∈g(X), let Pq,d,m(g, z) = {Πd ∈Mm,d : |g−1(Πd)| ≥ q and z ∈Πd}. There is a metric topology on Mm,d, see [6], and we consider Pq,d,m(g, z) as a subspace of Mm,d with this topology. One of the results from authors’ paper [4] states that if X is a metric compactum of dimension n and m ≥2n + 1, then the function space C(X, Rm) contains a dense Gδ-subset of maps g such that the set {Π1 ∈ Mm,1 : |g−1(Πd)| ≥2} is at most 2n-dimensional. The next theorem provides more information concerning the above result: Theorem 1.1. Let X be a metric compactum of dimension ≤n and m ≥2n + 1. Then the maps g : X →Rm such that dim P2,1,m(g, z) ≤0 for all z ̸∈g(X) form a dense Gδ-subset of C(X, Rm). Theorem 1.1 admits a parametric version. 1991 Mathematics Subject Classification. Primary 54C10; Secondary 54F45. Key words and phrases. compact spaces, algebraically independent sets, general position, dimension, Euclidean spaces. The first author was supported by Grants NSH 1562.2008.1 and RFFI 09-01- 00741-a. Research supported in part by NSERC Grant 261914-08. 1 2 Theorem 1.2. Let f : X →Y be a perfect n-dimensional map between metrizable spaces with dim Y = 0, and m ≥2n + 1. Then the maps g : X →Rm such that dim P2,1,m(g|f −1(y), z) ≤0 for all restrictions g|f −1(y), y ∈Y , and all z ̸∈g(f −1(y) form a dense Gδ-subset of C(X, Rm) equipped with the source limitation topology. For any map g ∈C(X, Rm) and z ̸∈g(X) we also consider the set D2,1,m(g, z) consisting of points y = (y1, y2) ∈(Rm)2 such that y1 and y2 belong to a line Π1 ⊂Rm with z ∈Π1, and there exist two different points x1, x2 ∈X with g(xi) = yi, i = 1, 2. Theorem 1.3 below follows from the proof of Theorem 1.2 by consid- ering the sets D2,1,m(g, z) instead of P2,1,m(g, z). Theorem 1.3. Let X, Y , f and m satisfy the hypotheses of Theorem 1.2. Then the maps g : X →Rm such that dim D2,1,m(g|f −1(y), z) ≤0 for all restrictions g|f −1(y), y ∈Y , and all z ̸∈g(f −1(y) form a dense Gδ-subset of C(X, Rm). Recall that for any metric space (M, ρ) the source limitation topol- ogy on C(X, M) can be describe as follows: the neighborhood base at a given function f ∈C(X, M) consists of the sets Bρ(f, ǫ) = {g ∈ C(X, M) : ρ(g, f) < ǫ}, where ǫ : X →(0, 1] is any continuous positive functions on X. The symbol ρ(f, g) < ǫ means that ρ(f(x), g(x)) < ǫ(x) for all x ∈X. It is well know that for metrizable spaces X this topology doesn’t depend on the metric ρ and it has the Baire property provided M is completely metrizable.
2. Preliminaries We need some preliminary information before proving Theorem 1.1. Everywhere in this section we suppose that q, m, d are integers with 0 ≤d ≤m and q ≥1. Moreover, the Euclidean space Rm is equipped with the standard norm ||.||m. We also suppose that X is a metric compactum and Γ = {B1, B2, .., Bq} is a disjoint family consisting of q closed subsets of X. For any g ∈C(X, Rm) and z ̸∈g(X) we denote by PΓ(g, z) the set {Πd ∈Mm,d : g−1(Πd) ∩Bi ̸= ∅for each i = 1, .., q and z ∈Πd}. Now, consider the open subset Rm X of C(X, Rm) × Rm consisting of all pairs (g, z) with z ̸∈g(X). Define the set-valued map ΦΓ : Rm X →Mm,d, ΦΓ(g, z) = PΓ(g, z). Proposition 2.1. ΦΓ is an upper semi-continuous and closed-valued map. Embeddings in Euclidean spaces 3 Proof. Suppose (g0, z0) ∈Rm X. We need to show that for any open W ⊂ Mm,d containing ΦΓ(g0, z0) there are neighborhoods O(g0) ⊂C(X, Rm) and O(z0) ⊂Rm with O(g0) × O(z0) ⊂Rm X and ΦΓ(g, z) ⊂W for all (g, z) ∈O(g0) × O(x0). Assume this is not true. Then there exists a sequence {(gk, zk)}k≥1 ∈Rm X converging to (g0, z0) and Πd k ∈PΓ(gk, zk) with Πd k ̸∈W, k ≥1. For any i ≤q and k ≥1 there exists a point xi k ∈Bi ∩g−1 k (Πd k). Since A = S i≤q g0(Bi) ⊂Rm is compact, we take a closed ball K in Rm with center the origin containing A in its interior. Because every Πd ∈PΓ(g0, z0) intersects A, we can identify PΓ(g0, z0) with {Πd ∩K : Πd ∈PΓ(g0, z0)} considered as a subspace of exp(K) (here exp(K) is the hyperspace of all compact subset of K equipped with the Vietoris topology). Because {gk}k≥1 converges in C(X, Rm) to g0, we can assume that K contains each set S i≤q gk(Bi), k ≥1. Hence, gk(xi k) ∈K ∩Πd k for all i ≤q and

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