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received (30 May 2026)

1 Introduction↩︎

L. Antoine constructed a Cantor set in plane such that all of its projections coincide with those of a regular hexagon [1]. Other examples with similar properties were described by Otto, A. Flores, G. Nöbeling [2]; see also [3], [4].

For each \(N\geqslant 2\), K. Borsuk constructed a Cantor set in \(\mathbb{R}^N\) such that its projection into any \((N-1)\)-plane contains a ball, equivalently, has dimension \((N-1)\) [5]. See also [6]. Many papers strengthen and extend Borsuk’s result, see e.g. [7], [8], [9], [10], [11].

In [12], J. Cobb described a Cantor set in \(\mathbb{R}^3\) such that its projection into any \(2\)-plane is \(1\)-dimensional, and posed a general question: for given integers \(N>m\geqslant k>0\), does there exist a Cantor set in \(\mathbb{R}^N\) such that its projection into any \(m\)-plane has topological dimension \(k\) ? (We call these sets briefly \((N,m,k)\)-sets.) The case of \(m=k\) is answered positively by Borsuk’s construction. For the cases \((N,m,k=m-1)\) and \((N, m=N-1 , k)\), desired sets were described in [13] and [14], correspondingly; both of these papers extend Cobb’s method. Applying facts from the theory of tame and wild Cantor sets, Frolkina obtained new wide series of \((N,N-1,N-1)\)- and \((N,N-1,N-2)\)-sets [4], [15].

Any zero-dimensional compactum in plane is tame [16], see also [17] or [18].

The first wild Cantor sets in \(\mathbb{R}^3\) were described by L. Antoine in 1920–21 and by P.S. Urysohn in 1922–1923 (independently). Antoine’s construction was later extended to \(\mathbb{R}^N\), \(N\geqslant 3\); now there are essentially different examples with additional properties. For references, see [19].

Using his \((N,N-1,N-1)\)-set, Borsuk constructed a simple arc in \(\mathbb{R}^N\) such that its projection into any \((N-1)\)-plane contains an \((N-1)\)-ball [5]. The same argument provides a knot in \(\mathbb{R}^N\) with the same property. For \(N=3\) this answers a question of R. Fox. Moreover: analyzing Borsuk’s paper, we see that a knot with this property exists in any equivalence class of knots (no matter, tame or wild). See Proposition 1.

Compacta which are embedded in a topologically equivalent way may behave absolutely different if we consider dimensions of their projections. Proposition 1 includes several statements of this type. (For a tame Cantor set, compare 1a) and 1b). For a wild Cantor set, compare 2) and 3). For a knot, see 3) and 4).)

The aim of this paper is to understand which behavior of isotopies is typical for a given compactum \(X\subset\mathbb{R}^N\). “Typical” is understood in the sense of Baire category. Below we answer this question. At the same time, we strengthen Väisälä’s theorem [21].

Formally, Proposition 1 is new. Its parts 1a), 3) strengthen the result from [5] and from [4]. Statement 1b) implies [22], [12]. Part 2) implies [4]. Part 4) implies [5]. But main ideas which prove Proposition 1 are contained in [5], [4]. Proposition 1 is stated here to motivate the main problem. We prove it in Section 5.1. Part 4) is strengthened in Corollary 2.

Similar theorems about projections exist in measure theory: instead of topological dimension, the Hausdorff dimension is considered, and projections are taken on “almost all" planes, i.e., with the exception of the set of planes with measure zero. See [23], [24], [3].

1.1 Agreements, notation↩︎

\(\overline{A}\) and \(\partial A\) are the closure and the boundary of a set \(A\), correspondingly.

\(\dim X\) denotes topological dimension of a separable metric space \(X\) [25], [26] (for such spaces, three classical topological definitions of dimension are equivalent [26]).

\(m_q (X)\) is the \(q\)-dimensional Hausdorff measure, and \(\dim _H X\) is the Hausdorff dimension of a space \(X\). See [25].

A linear subspace \(\Pi \subset \mathbb{R}^N\) is proper if \(\{ 0 \} \neq \Pi \neq \mathbb{R}^N\).

2 About the concepts needed to formulate an answer↩︎

2.1 Embedding dimension↩︎

We need Shtan’ko’s “embedding dimension" theory.

The notation “dem” is an abbreviation for “dimension of embedding”. We have \(\dim X \leqslant \operatorname{dem}X\). This number is not a topological invariant of \(X\), however, it is an invariant of the equivalence class of the embedding \(X\hookrightarrow \mathbb{R}^N\). This means that \(\operatorname{dem}X = \operatorname{dem}h(X)\) for any homeomorphism \(h:\mathbb{R}^N\cong\mathbb{R}^N\) [27]. Briefly, for \(\operatorname{dem}X = k\), the embedded compactum \(X\) behaves “like” a \(k\)-dimensional polyhedron. This concept was introduced by Stan’ko with the aim of extending the definition of tame and wild embeddings to arbitrary compact sets. Previously, definitions of tameness and wildness were available only for embeddings of polyhedra and zero-dimensional compacta.

For a Cantor set \(K\subset \mathbb{R}^N\), tameness is equivalent to the equality \(\operatorname{dem}K = 0\) [27], [17].

The demension of each wild Cantor set in \(\mathbb{R}^N\) equals \(N-2\) [28], [27], [29], [30] (for \(N=4\), see also [31], [32], [33], [34], [35]).

The case of \(1\)-dimensional compacta in \(\mathbb{R}^3\) is special. In particular, any embedding of a segment or circle in \(\mathbb{R}^3\) is tame in the sense of Shtan’ko, i.e., its embedding dimension equals \(1\) [36] .

For details (in particular, other equivalent definitions of the dimension \(\operatorname{dem}\)) see [27], [29], [30].

2.2 Theorems of Szpilrajn and Väisälä↩︎

In [37], Nöbeling proved that \[\dim X \leqslant \dim _H X .\] In [38] (see also [25]), Szpilrajn strengthened this showing that for a separable metric space \(X\) \[\text{ if } m_{n+1}(X) =0, \text{ then } \dim X \leqslant n.\] Moreover, for \(\dim X \leqslant n\) a typical continuous map \(f \in C(X, I^{2n+1})\) satisfies \(m_q (f(X)) = 0\) for any \(q >n\), hence \(\dim _H f(X) \leqslant n\) [38], [25]. The latter fact strengthens the classical Lefschetz–Menger–Nöbeling–Pontryagin–Tolstowa embedding theorem [25]. It also implies that \(\dim X = \inf \{ \dim_H Y\},\) where \(Y\) runs over all spaces homeomorphic to \(X\) [38].

For any compactum \(X\subset \mathbb{R}^N\) we have (see [30] and [39]) \[\dim X \leqslant \operatorname{dem}X\leqslant \dim_H X .\]

J. Väisälä proved an analogue of Szpilrajn’s theorem for \(\operatorname{dem}\). He showed that for a compactum \(X\subset \mathbb{R}^N\) the inequality \(\operatorname{dem}X \leqslant k\) is equivalent to the existence of a homeomorphism \(f:\mathbb{R}^N\cong\mathbb{R}^N\) with \(m_{k+1}(f(X)) = 0\). Moreover, \(\operatorname{dem}X \leqslant \dim _H (f(X))\) for each homeomorphism \(f:\mathbb{R}^N\cong\mathbb{R}^N\), and this becomes an equality for some homeomorphism \(f\) [21], [30]. We can also control the size of the isotopy and its support [21].

3 Statements↩︎

All proofs are in Section 5.

Theorem 2 is the main result of this paper. It simultaneously strengthens Väisälä’s theorem and provides an answer to the question posed in Section 1 about the typical behavior of projections of compact sets.

On the notion of “typical” see Section 4.1.

Spaces \(\operatorname{Homeo}_\varepsilon (\overline{U},\partial U)\) and \(\operatorname{Isot}_\varepsilon (\overline{U},\partial U)\) are defined in Section 4.3.

This immediately implies

To prove the existence of the desired homeomorphism \(f\), Väisälä uses the equivalence of the inequality \(\operatorname{dem}X \leqslant k\) and of the ambient embeddability of \(X\) into the Menger compactum \(M^k_N\) [40], see also [29], [30]. Väisälä achieves the control over the isotopy using Edwards’s remark [29]: for a given \(\varepsilon\), instead of the standard Menger compactum one should take a more complicated subset, depending on \(\varepsilon\). Our work does not use Menger compacta, and the proofs rely solely on the definition of the embedding dimension \(\operatorname{dem}\), standard properties of the Hausdorff measure, and the Baire category.

Now we can find out how often a knot in \(\mathbb{R}^3\) has two-dimensional projections, i.e., how “common” the Borsuk-type property is. Recall that a typical knot in \(\mathbb{R}^3\) is wild [41] and even wild at every point [42].

Corollaries 3, 4 include similar statements for Cantor sets.

In the case of zero-dimensional compact sets, the use of “double control” (by the \(\varepsilon\) and by the support \(U\)) is redundant.

4 Necessary concepts↩︎

4.1 Baire category↩︎

For this definition to make sense, we restrict ourselves to considering sec:Baire spaces (i.e., spaces in which every nonempty open subset has second category in \(\mathcal{X}\); equivalently, every residual subset is dense in \(\mathcal{X}\).)

A space metrizable by a complete metric is a Baire space (R. Baire for \(\mathbb{R}\), 1899; F. Hausdorff, 1914). A \(G_\delta\)-subset of a space metrizable by a complete metric can also be metrized by a complete metric by P.S. Alexandroff’s theorem [43].

4.2 Spaces of compacta↩︎

To illustrate Section 4.1, we remind some well-known examples; they will be used below.

4.3 Spaces of homeomorphisms and isotopies↩︎

Let \((Y, \rho )\)be a non-empty metric compactum, and \((\widetilde{Y}, \widetilde{\rho} )\) a metric space. The space of continuous maps \(C(Y,\widetilde{Y})\) is endowed with the metric \[d(f,g) = \sup\limits _{x\in Y} \rho (f(x),g(x)).\] If \(\widetilde{Y}\) is Polish, then \(C(Y,\widetilde{Y})\) is also Polish [43]. In this paper, \(\widetilde{Y}\) either coincides with \(Y\), or is a Euclidean space.

We will use the following subspaces of \(C(Y,Y)\):

1) \(\operatorname{Homeo}(Y)\) — the set of all homeomorphisms \(Y\cong Y\).

2) \(\operatorname{Homeo}(Y,A)\) — the set of all homeomorphisms \(f:Y\cong Y\) such that⁶ \({f|_A = \operatorname{id}}\).

3) \(\operatorname{Homeo}_{\varepsilon }(Y,A)\), where \(\varepsilon >0\), is the set of all homeomorphisms \(f:Y\cong Y\) such that \(f|_A = \operatorname{id}\) and \(d(f,\operatorname{id}) < \varepsilon\).

For \(\varepsilon > \operatorname{diam}Y\) we obviously have \(\operatorname{Homeo}_{\varepsilon }(Y,A)=\operatorname{Homeo}(Y,A)\).

We will use the following subspaces of \(C(Y\times I, Y)\):

4) \(\operatorname{Isot}(Y)\) is the set of all isotopies of \(Y\). The distance between \(F=\{ f_t \} ,G=\{ g_t\} \in\operatorname{Isot}(Y)\) is defined by \[D(F,G) = \sup _{x\in Y , t\in I} \rho (f_t (x) , g_t (x)) .\]

5) \(\operatorname{Isot}(Y,A)\) is the set of all isotopies \(F:Y\times I\to Y\) such that \(f_t |_A = \operatorname{id}\) for each \(t\in I\).

6) \(\operatorname{Isot}_{\varepsilon } (Y,A)\), where \(\varepsilon >0\), is the set of all isotopies \(F:Y\times I\to Y\) such that⁷ \(f_t |_A = \operatorname{id}\) for each \(t\in I\) and \(D (F , \operatorname{Id}) < \varepsilon\).

Hence each of the spaces \(\operatorname{Homeo}(Y)\), \(\operatorname{Homeo}(Y,A)\), \(\operatorname{Homeo}_{\varepsilon }(Y,A)\), \(\operatorname{Isot}(Y)\), \(\operatorname{Isot}(Y,A)\), \(\operatorname{Isot}_{\varepsilon }(Y,A)\) is completely mertizable [43]. When establishing the openness or everywhere density of subsets of these spaces, we will use “standard, uniform” metrics \(d\), \(D\) induced from \(C(Y,Y)\) and \(C(Y\times I,Y)\). These metrics are in general incomplete. This is permissible, since the indicated properties are topological and can be verified in any of the metrics defining the same topology.

5 Proofs↩︎

5.1 Proof of Proposition 1↩︎

1a) Is a variation of Statement 12 from [4]. For completeness, we give a proof.

First consider a particular case: \(X\) is a Cantor \((N,N-1,N-1)\)-set obtained by the construction of Borsuk [5]. There is quite a lot of “freedom” in this construction, however, each of these sets is tame according to the Bing–Keldysh–Osborne tameness criterion [17] (see also [4]).

In fact, Borsuk’s construction shows that for any non-empty open subset \(A\subset X\) we have: \(\dim P_\Pi (A) = N-1\) for any \((N-1)\)-plane \(\Pi\). Thus in the particular case under consideration we may take \(h_t = \operatorname{id}\) for each \(t\in I\).

Now let \(X\) be an arbitrary tame Cantor set. Let \(\{ L_i\}\) be a defining sequence for \(X\) [4]. There is a positive integer \(k\) with the property: the diameter of each connected component \(L_k^{(1)},\ldots , L_k^{(s)}\) of \(L_k\) is smaller than \(\varepsilon\). We may assume that the intersection \(X_i:= X\cap L_k^{(i)}\) is non-empty for each \(i = 1,\ldots , s\) (hence all \(X_i\)’s are tame Cantor sets). It is clear that Borsuk’s construction provides Cantor sets of arbitrarily small diameter (apply a homothety is necessary). Hence for each \(i = 1,\ldots , s\) there exists a tame Cantor \((N,N-1, N-1)\)-set \(Y_i \subset \operatorname{Int}L_k^{(i)}\) [5]. For any non-empty open subset \(A\subset Y_1\cup \ldots \cup Y_s\) the projection of \(A\) into any \((N-1)\)-plane has dimension \((N-1)\). The proof of Theorem I.4.2 from [17] (see also [4]) implies: there is an isotopy \(\{ h_t \} : \mathbb{R}^N\cong \mathbb{R}^N\) such that \(h_t |_{\mathbb{R}^N - \cup _{i=1}^s L_k^{(i)} } = \operatorname{id}\) for each \(t\in I\), and \(h_1(X_i) = Y_i\) for each \(i=1,\ldots , s\). This is the desired isotopy.

1b) is proved similarly to 1a). Now \(Y_i\)’s are choosed differently: they should be tame Cantor sets in general position with respect to all projections [12], [20], [48]. The union \(Y:= Y_1\cup \ldots \cup Y_s\) is a Cantor set [43] and it is tame [6], [49]. Moreover, \(Y_i\)’s can be taken so that \(Y\) has general position with respect to all projections.

2) is proved similarly to Theorem 2 from [4].

3) Any non-countable compactum \(X\) contains a Cantor set \(K\) [43]. Moreover: if \(X\) is embedded in \(\mathbb{R}^N\), then \(K\) can be chosen to be tame in \(\mathbb{R}^N\). (For this, use the tameness criterion [17], [4].) To finish the proof, apply 1a).

4) By [36], \(\operatorname{dem}X =1\). Together with [21] this implies: there is an \(\varepsilon\)-isotopy \(\{ h_t \} : \mathbb{R}^3 \cong \mathbb{R}^3\) with support in \(O_\varepsilon X\) such that \(\dim _H h_1( X) = 1\). Orthogonal projection can not raise Hausdorff dimension; hence \(\dim _H P_\Pi (h_1( X) ) \leqslant 1\) for any linear subspace \(\Pi \subset \mathbb{R}^3\). For a non-zero subspace \(\Pi \subset \mathbb{R}^3\) the equality \(\dim _H P_\Pi (h_1 (X)) = 0\) is possible only in two cases:

(i) \(\dim \Pi = 2\), and \(h_1(X)\) is a segment perpendicular to \(\Pi\);

(ii) \(\dim \Pi = 1\), and \(h_1(X)\) is contained in a plane perpendicular to \(\Pi\).

In both cases, it is easy to “tweak” the isotopy \(\{ h_t \}\) to obtain the desired property.

5.2 Proof of Proposition 4↩︎

1) It is known that \(\operatorname{Homeo}(Y)\) is a \(G_\delta\)-subset of \(C(Y,Y)\) [30]. Observe that \[\operatorname{Homeo}(Y,A) = \operatorname{Homeo}(Y) \cap \{ f\in C(Y,Y) \;: \;f|_A = \operatorname{id} \} .\] The set \(\{ f\in C(Y,Y) \;: \;f|_A = \operatorname{id} \}\) is closed in \(C(Y,Y)\), hence is a \(G_\delta\)-subset of \(C(Y,Y)\). Consequently \(\operatorname{Homeo}(Y,A)\) is a \(G_\delta\)-subset of \(C(Y,Y)\).

We further observe that \[\operatorname{Homeo}_{\varepsilon } (Y,A) = \operatorname{Homeo}(Y,A) \cap \{ f\in C(Y,Y) \;: \;d (f, \operatorname{id}) < \varepsilon \} .\] The second of the two intersected sets is open in \(C(Y,Y)\). This proves 1). Part 2) can be proved by similar reasoning.

5.3 Proof of Theorem 2↩︎

The proof is very close to the arguments of S. Eilenberg, see the proof of Theorem 3 in [38] or the proof of Theorem VII 5 in [25].

a) For \(q>0\) and \(k\in\mathbb{N}\) define \(W_{q , k }\) as the set of all \(f\in \operatorname{Homeo}_\varepsilon (\overline{U} , \partial U)\) with the property: for some finite collection \(V_1, \ldots ,V_s\) of open subsets of \(\mathbb{R}^N\) we have \[f(X) \subset V_1 \cup \ldots \cup V_s \subset U \quad \text{ and } \quad \sum_{i=1}^{s} (\operatorname{diam}V_i)^q < \frac{1}{k} .\] Evidently each \(W_{q , k }\) is open in \(\operatorname{Homeo}_\varepsilon (\overline{U} , \partial U)\). Hence the set \[T := \bigcap\limits_{\substack{q= \operatorname{dem}X +\frac{1}{n}\\ n\in \mathbb{N}}} \left( \bigcap\limits_{k \in \mathbb{N}} W_{q, k } \right)\] is a \(G_\delta\)-subset of \(\operatorname{Homeo}_\varepsilon (\overline{U} , \partial U)\).

For each \(f\in T\) we have \(\dim _H f(X) = \operatorname{dem}X\). In fact, take any \(f\in T\). For each \(q >\operatorname{dem}X\) we have \({m_q (f(X)) = 0}\) by [25] and by definition of \(T\). Hence \(\dim _H f(X) \leqslant \operatorname{dem}X\). Recall that \(\operatorname{dem}\) is invarint under homeomorphisms of \(\mathbb{R}^N\). We get \(\operatorname{dem}X = \operatorname{dem}f(X)\) [27]. This together with the inequality \(\operatorname{dem}f(X) \leqslant \dim _H f(X)\) [39], [30] implies \(\dim _H f(X) = \operatorname{dem}X\).

Fix any \(q >\operatorname{dem}X\) and any \(k \in\mathbb{N}\). Let us show that the set \(W_{q , k }\) is dense in \(\operatorname{Homeo}_\varepsilon (\overline{U} , \partial U)\). To show this, take any \(g\in \operatorname{Homeo}_\varepsilon (\overline{U} , \partial U)\) and \(\gamma >0\). Let us construct an \(f\in W_{q, k }\) with \(d(f,g) < \gamma\). Take a positive number \[\delta < \min \Bigl\{ \gamma ; \quad \frac{1}{3} d (g(X) , \partial U); \quad \varepsilon - d(g,\operatorname{id}) \Bigr\} .\] Since \(\operatorname{dem}g(X) = \operatorname{dem}X\), there exists a finite polyhedron \(P\subset \mathbb{R}^N\) of dimension \(\dim P = \operatorname{dem}X\) and a \(\delta\)-pseudoisotopy \(\Phi = \{ \varphi _t \} : \mathbb{R}^N\to \mathbb{R}^N\) with support in \(O_{\delta } (g(X))\) such that \(\varphi _1 (g(X)) = P\) [27]. Note that \(O_{\delta } P \subset O_{2\delta } (g(X)) \subset U\).

We have \(m_q P = 0\) since \(q >\operatorname{dem}X = \dim P\). Hence there is a representation \(P=L_1 \cup \ldots \cup L_r\) with \[\sum\limits_{i=1}^{r} (\operatorname{diam}L_i)^q < \frac{1}{k} .\] Let \(\widetilde{L}_1, \ldots , \widetilde{L}_r\) be open subsets of \(\mathbb{R}^N\) with \[\sum \limits_{i=1}^r (\operatorname{diam}\widetilde{L}_i)^q < \frac{1}{k}, \quad \bigcup \limits_{i=1}^r \widetilde{L}_i \subset U, \quad L_i \subset \widetilde{L}_i \text{ for each } i=1,\ldots ,r.\] Continuity of \(\Phi : \mathbb{R}^N \times I\to\mathbb{R}^N\) and compactness of \(g(X)\) imply: \(\varphi _\tau (g(X)) \subset \bigcup\limits _{i=1}^r \widetilde{L}_i\) for some \(\tau \in [0,1)\).

Define \(f:= \varphi _\tau \circ g\). Then \[d(f,g) = d(\varphi _\tau \circ g ,g) < d(\varphi _\tau , \operatorname{id}) \leqslant \delta <\gamma\] and \[d(f,\operatorname{id})\leqslant d(f,g)+d(g,\operatorname{id}) < \delta + \left( \varepsilon - \delta \right) =\varepsilon .\] Together with \[f(X) = \varphi _\tau (g(X)) \subset \bigcup\limits _{i=1}^r \widetilde{L}_i \subset U \quad \text{ and } \quad \sum \limits_{i=1}^r (\operatorname{diam}\widetilde{L}_i)^q < \frac{1}{k}\] this implies: \(f\in W_{q , k }\). Thus a) is proved.

Part b) is proved by analogous reasoning. For \(q >0\) and \(k\in\mathbb{N}\) instead of \(W_{q, k}\) consider \(\mathbb{W}_{q , k }\) consisting of all \(F\in \operatorname{Isot}_\varepsilon (\overline{U}, \partial U)\) with the property: for some finite collection \(V_1, \ldots ,V_s\) of open subsets of \(\mathbb{R}^N\) we have \[f_1 (X) \subset V_1 \cup \ldots \cup V_s \subset U \quad \text{ and } \quad\sum_{i=1}^{s} (\operatorname{diam}V_i)^q < \frac{1}{k} .\] We similarly define \[\mathbb{T} := \bigcap\limits_{\substack{q = \operatorname{dem}X+\frac{1}{n}\\ n\in \mathbb{N}}} \left( \bigcap\limits_{k \in \mathbb{N}} \mathbb{W}_{q , k } \right) ,\] this is a \(G_\delta\)-subset of \(\operatorname{Isot}_\varepsilon (\overline{U}, \partial U)\).

The density of \(\mathbb{W}_{q, k}\) in \(\operatorname{Isot}_\varepsilon (\overline{U}, \partial U)\) for \(q >\operatorname{dem}X\) and \(k \in \mathbb{N}\) is somewhat more involved than for part a), but is not difficult. Briefly, given an isotopy \(G=\{ g_t \} \in \operatorname{Isot}_\varepsilon (\overline{U}, \partial U)\) and a number \(\gamma >0\), then constructing an isotopy \(F=\{ f_t \} \in \mathbb{W}_{q, k}\) satisfying \(D(F,G) < \gamma\) can be done as follows: successively apply the isotopy \(G\), and then some isotopy obtained by incompletely applying a small pseudoisotopy, mapping \(g_1 (X)\) to a polyhedron of dimension equal to \(\operatorname{dem}X\).

5.4 Proof of Corollary 2↩︎

a) Consider two cases.

A particular case. The set \(\Sigma \subset \mathbb{R}^3\) is embedded equivalently to a standard circle in \(\mathbb{R}^2 \times \{ 0\}\) or a segment in \(\mathbb{R} \times \{ 0\}\times \{ 0\}\). In this case, a) is easy. Moreover: the set of all homeomorphisms \(f\in \operatorname{Homeo}_\varepsilon (\overline{U} , \partial U)\) such that \(f (\Sigma )\) lies in a \(2\)-plane is nowhere dense in \(\operatorname{Homeo}_\varepsilon (\overline{U} , \partial U)\).

The general case. The embedding \(\Sigma \subset \mathbb{R}^3\) is non-equivalent to a standard circle or a segment. Obviously \(\dim P_{\Pi } \Sigma \neq 0\) for any \(2\)-plane \(\Pi\).

Let us show that \(\dim P_{\ell } \Sigma \neq 0\) for any straight line \(\ell\). Suppose the contrary. Then \(\Sigma \subset L\), where \(L\) is a \(2\)-plane perpendicular to \(\ell\). By Jordan–Schönflies Theorem [18], there is a homeomorphism \(h:L\cong L\) such that \(h(\Sigma )\) is a circle or a segment. Hence \(H= h\times \operatorname{id}: L\times \ell \cong L\times \ell\) is a homeomorphism of \(\mathbb{R}^3\) which takes \(\Sigma\) into a circle or a segment, a contradiction.

Recall that \(\operatorname{dem}\Sigma = 1\) for any knot and any simple arc \(\Sigma \subset \mathbb{R}^3\) [36]. This together with Theorem 2 implies: \(\dim _H f(\Sigma ) = 1\) for a typical homeomorphism \(f\in \operatorname{Homeo}_\varepsilon (\overline{U} , \partial U)\). For any \(2\)-plane or line \(\Pi \subset \mathbb{R}^3\) we then get \[0\neq \dim P_\Pi (f(\Sigma )) \leqslant \dim _H P_\Pi (f(\Sigma )) \leqslant \dim _H f(\Sigma ) = 1.\] Hence \(\dim P_\Pi (f(\Sigma )) =1\), and a) is proved.

Similar arguments prove Part b).

5.5 Proof of Corollary 3↩︎

\((i) \Rightarrow (iii)\). For any tame Cantor set \(K\subset \mathbb{R}^N\) we have \(\operatorname{dem}K = 0\). By Theorem 2, \(\dim _H f(X) =0\) for a typical element \(f\in \operatorname{Homeo}(\overline{U} , \partial U)\). Hausdorff dimension can not raise under projections. Hence \[\dim P_\Pi (f (K)) \leqslant \dim _H P_\Pi (f (K)) \leqslant \dim _H f (K) = 0\] for any linear subspace \(\Pi \subset \mathbb{R}^N\).

\((i) \Rightarrow (iv)\) is proved by similar arguments.

It is evident that any of \((iii), (iv), (v)\) implies \((ii)\).

Let us prove that \((ii)\Rightarrow (i)\).

Case 1. For \(N=2\) there is nothing to prove.

Case 2. Let \(N=3\), and \(\dim P_\Pi (f(K)) = 0\) for some \(2\)-plane \(\Pi\). We may assume that \(\Pi = \mathbb{R} \times \mathbb{R} \times \{ 0 \}\) (a coordinate plane). By Antoine’s theorem, any zero-dimensional compactum in plane is tame [16], [17], [18]. Hence there is a homeomorphism \(h:\Pi \cong \Pi\) with \(h( P_\Pi (f(K))) \subset \mathbb{R} \times \{ 0\}\). Define a homeomorphism \(\widetilde{h} = h\times \operatorname{id}_\mathbb{R} : \mathbb{R}^3 \cong\mathbb{R}^3\), we have \(\widetilde{h} (f(K)) \subset \mathbb{R} \times \{0\}\times \mathbb{R}\). Applying Antoine’s theorem again, we get: \(f(K)\) is tame in \(\mathbb{R}^3\). Consequently \(K\) is tame.

Case 3. \(N\geqslant 3\), and \(\dim P_\Pi (f(K)) = 0\) for some straight line \(\Pi\). Necessary statement follows from [4].

Case 4 (the general one). \(N\geqslant 4\), and \(\dim P_\Pi (f(K)) = 0\) for some linear subspace \(\Pi\) with \(2\leqslant \dim \Pi \leqslant N-1\). Let \(\widetilde{\Pi} \subset \mathbb{R}^N\) be an \((N-1)\)-dimensional linear subspace containing \(\Pi\). We have \(\dim P_{\widetilde{\Pi} } (f(K)) \leqslant N-1-\dim \Pi \leqslant N-3\), and desired statement follows from [50], [51]. (These two papers use the fact: a Cantor set \(X\subset \mathbb{R}^N\) is tame iff \(\mathbb{R}^N - X\) is \(1\)-LC, see [30]. For the cases \(\dim \Pi = N-1\) or \(\dim \Pi = N-2\), a more direct proof of tameness of \(K\) is given in [4], where Bing–Keldysh–Osborne tameness criterion is used.)

Finally, we show that \((i)\Rightarrow (v)\). For a non-empty open set \(U\subset \mathbb{R}^N\) by \(\mathcal{T}'(U)\) denote the space of all tame Cantor sets \(Y\subset \mathbb{R}^N\) which have non-empty intersection with any connected component of \(U\). The space \(\mathcal{T}'(U)\) is endowed with the Hausdorff metric. It is completely metrizable, see Proposition 3. By [43] together with Proposition 4, the space \(\operatorname{Homeo}( \overline{U} , \partial U)\) is also completely mertizable. The proof in [17] (see also [4]) implies that the group action \[\Psi : \operatorname{Homeo}( \overline{U} , \partial U) \times \mathcal{T}'(U) \to \mathcal{T}'(U), \quad (h,X)\mapsto h(X) .\] is transitive. By Effros Theorem [52], [53], [54], the map \[\Psi _K : \operatorname{Homeo}( \overline{U} , \partial U) \to \mathcal{T}'(U), \quad h\mapsto h(K)\] is an open surjection. Now \((v)\) follows since a typical element of \(\mathcal{T}'(U)\) is a Cantor set having general position with respect to all projections [20], [48].

5.6 Proof of Corollary 4↩︎

Implications \((v), (vi), (vii)\Rightarrow (i)\iff (ii)\) follow from Corollary 3. Implications \((ii)\Rightarrow (iii)\Rightarrow (i)\) and \((iv)\Rightarrow (i)\) are evident. (For \((i)\Rightarrow (iii)\) see also [4].)

\((i)\Rightarrow (iv)\). For \(N=3\) see the proof of Corollary 3, namely, of \((ii)\Rightarrow (i)\). Now let \(N\geqslant 4\). Suppose the contrary: for some homeomorphism \(f:\mathbb{R}^N\cong\mathbb{R}^N\) and some proper linear subspace \(\Pi\) we have \(\dim P_\Pi (f(K)) \leqslant \dim \Pi -2\). Take an \((N-1)\)-dimensional linear subspace \(\tilde{\Pi} \supset \Pi\). Then \[\dim P_\Pi (f(K)) \leqslant ( \dim \Pi -2) + (\dim \tilde{\Pi} - \dim \Pi ) = N-3.\] By [50], [51], \(f(K)\) is tame. Consequently \(K\) is tame.

Let us prove \((i)\Rightarrow (v)\). For any wild Cantor set \(K\subset \mathbb{R}^N\) we have \(\operatorname{dem}K = N-2\). Hence \(\dim _H f(X) =N-2\) for a typical element \(f\in \operatorname{Homeo}(\overline{U} , \partial U)\). Then for any \((N-1)\)-plane \(\Pi \subset \mathbb{R}^N\) we have \[\dim P_\Pi (f (K)) \leqslant \dim _H P_\Pi (f (K)) \leqslant \dim _H (f (K)) = N-2 .\]

Suppose that \(\dim P_\Pi (f (K)) \leqslant N-3\) for some \((N-1)\)-plane \(\Pi\). Then \(f(K)\) is tame [50], [4]. Consequently \(K\) is tame, a contradiction. Thus \(\dim P_\Pi (f (K)) = N-2\) for any \((N-1)\)-plane \(\Pi\).

The proof of \((i)\Rightarrow (vii)\) is similar to that of \((i)\Rightarrow (v)\).

Finally, we show that \((i)\Rightarrow (vi)\). As above: for a typical element \(f\in \operatorname{Homeo}(\overline{U} , \partial U)\) we have \(\dim _H f(X) =N-2\). To get necessary statement, apply [24] (see p. 105–106 of the cited book for historic details and references).

References↩︎