MATHEMATICS

Lyudmila KELDYSH

EMBEDDING OF ZERO-DIMENSIONAL COMPACTA IN \(E^n\)

(Presented by Academician P. S. Aleksandrov, 19 VI 1962)

1. A zero-dimensional compactum \(A \subset E^n\), where \(E^n\) is \(n\)-dimensional Euclidean space, can be “wildly” embedded in \(E^n\) if \(n \geqslant 3\), i.e., in such a way that there does not exist a homeomorphic mapping \(f : E^n \to E^n\) for which \(f(A) \subset E^1\), where \(E^1\) is a line in \(E^n\) \((^{1,2})\). A zero-dimensional compactum \(A \subset E^n\) can be represented in the form

\[ A=\bigcap_{k=1}^{\infty}\bigcup_{i=1}^{n_k} V_i^k;\qquad \overline{V}_i^k\cap\overline{V}_j^k=\Lambda,\quad i\ne j;\qquad \bigcup_{i=1}^{n_k}\overline{V}_i^k\subset \bigcup_{j=1}^{n_{k-1}} V_j^{k-1}, \tag{1} \]

where \(V_i^k\) are domains in \(E^n\), whose diameters tend to zero as \(k\) increases, and \(\overline{\mathscr E}\) is the closure of the set \(\mathscr E\). It follows from (3) that if \(n=3\) and all \(\overline{V}_i^k\) are topological balls, then \(A\) is tamely embedded in \(E^3\), i.e., there exists a homeomorphism \(f : E^3 \to E^3\) such that \(f(A)\subset E^1\). We shall show that this holds for any \(n\), and shall consider the question of obtaining an arbitrary zero-dimensional compactum in \(E^n\) by means of a homotopy \(F_t : E^n \to E^n\), for every \(t\), \(a \leqslant t \leqslant b\), starting from a compactum lying on a line, and the identity mapping \(F_0\).

Definition 1. A compactum \(A \subset E^n\) is called cellularly separated in \(E^n\) if, in any of its neighborhoods \(V\), one can inscribe a neighborhood \(U\), the closure of which is the sum of a finite number of pairwise disjoint \(n\)-dimensional topological balls (\(n\)-elements):

\[ A\subset U\subset \overline U\subset V;\qquad U=\bigcup_{r=1}^{m} U_r;\quad \overline U_r\cap\overline U_{r'}=\Lambda,\quad r\ne r';\quad \overline U_r\text{ is an }n\text{-element}. \]

Definition 2. A pseudo-isotopy \(\Phi_t,\ 0\leqslant t\leqslant 1\), of a space \(X\) onto itself is a homotopy taking the identity mapping \(\Phi_0\) into a continuous mapping \(\Phi_1:X\to X\) such that, for every \(t<1\), \(\Phi_t\) is a homeomorphic mapping of \(X\) onto itself.

Theorem 1. For every cellularly separated zero-dimensional compactum \(A\) in \(E^n\) there exists an isotopy \(F_t:E^n\to E^n,\ 0\leqslant t\leqslant 1\), from the identity mapping \(F_0\) such that \(F_1(A)\) lies on a line.

Theorem 2. For an arbitrary zero-dimensional compactum \(A\) in \(E^n\) there exists a compactum \(C\), lying on a line, and a pseudo-isotopy \(\Phi_t,\ 0\leqslant t\leqslant 1\), of the space \(E^n\) onto itself such that \(\Phi_1(C)=A\), and the mapping \(\Phi_1\) is a homeomorphism on the set \(E^n\setminus \Phi_1^{-1}(A)\) and on the compactum \(C\).

In the case when \(A\) has no isolated points, the Cantor perfect set may be taken as \(C\).

2. Lemmas. For all the isotopies considered below, \(F_0\) is the identity mapping; therefore we shall not stipulate this.

Lemma 1. Let \(\delta^n,\delta'^n\) be two closed balls in \(E^n\) with centers \(O,O'\) and radii \(r,r'\), and let \(U\) be a domain containing \(\delta^n\cup\delta'^n\). There exist-

there is an isotopy \(F_t:E^n\to E^n\) from the identity \(F_0\) such that \(F_1(\delta^n)=\delta^{\prime n}\) and \(F_t(x)=x\), if \(x\in E^n\setminus U\), \(0\leq t\leq 1\).

If \(O=O'\), then there exists an isotopy \(F_t\) carrying linearly each radius of the ball \(\delta^n\) into a radius of the ball \(\delta^{\prime n}\) and fixed outside an arbitrary ball concentric with \(\delta^n\) and \(\delta^{\prime n}\) and containing \(\delta^n\cup\delta^{\prime n}\).

If the points \(O\) and \(O'\) are distinct, then they can be joined in \(U\) by a polygonal line

\[ l=\bigcup_{i=1}^{k} l_i, \]

where \(l_i\) are the links of \(l\). For each \(l_i\) choose a cylindrical neighborhood \(C_i\) so that

\[ l_i\subset C_i\subset \overline C_i\subset U;\qquad \overline C_i\cap \overline C_j=\Lambda,\quad \text{if } |i-j|>1. \]

Each vertex \(a_i=l_i\cap l_{i+1}\) is contained in \(C_i\cap C_{i+1}\). Choose a number \(\rho\) such that for every \(i\) the ball \(\delta_i\) of radius \(\rho\) with center at the point \(a_i\) is contained in \(C_i\cap C_{i+1}\). First construct an isotopy \(F_t\), \(0\leq t\leq 1/(k+2)\), so that \(F_0(x)\equiv x\), \(F_{1/(k+2)}(\delta^{\prime n})=\delta_0^n\), where \(\delta_0^n\) is the ball of radius \(\rho\) with center \(O\), and \(F_t(x)=x\) outside a small neighborhood \(\delta^{\prime n}\cup \delta_0^n\).

For each \(i\) an isotopy \(\varphi_t^i\) is constructed for \(i/(k+2)\leq t\leq (i+1)/(k+2)\), piecewise linearly carrying into itself every straight line parallel to \(l_i\), such that \(\varphi_t^i(x)=x\), if \(x\in E^n\setminus C_i\); \(\varphi_{(i+1)/(k+2)}^i(\delta_{i-1}^n)=\delta_i^n\); \(\varphi_{i/(k+2)}^i(x)\equiv x\). Finally, for \((k+1)/(k+2)\leq t\leq 1\) the isotopy \(\varphi_t^{k+1}\) is fixed outside a small neighborhood \(\delta^{\prime n}\cup \delta_k^n\), and \(\varphi_1^{k+1}(\delta_k^n)=\delta^{\prime n}\).

The isotopy \(F_t\) is defined successively by the equalities

\[ F_t(x)=\varphi_t^i F_{i/(k+2)}(x),\qquad i/(k+2)\leq t\leq (i+1)/(k+2),\quad i=1,2,\ldots,k+1. \]

Since all \(\varphi_t^i\) are fixed outside \(U\), we also have \(F_t(x)=x\), if \(x\in E^n\setminus U\), \(0\leq t\leq 1\). By construction \(F_1(\delta^n)=\delta^{\prime n}\).

Denote by \(\operatorname{int}\mathscr E\) the set of interior points of \(\mathscr E\), and \(\partial\mathscr E=\overline{\mathscr E}\setminus \operatorname{int}\mathscr E\).

Lemma 2. For any \(n\)-element \(Q^n\subset E^n\), compactum \(K\subset \operatorname{int} Q^n\), ball \(\delta^n\), and domain \(U\supset Q^n\cup \delta^n\), there exist an \(n\)-element \(Q^{\prime n}\subset \operatorname{int} Q^n\), containing \(K\), and an isotopy \(F_t\) from the identity \(F_0\), fixed on \(E^n\setminus U\), such that \(F_1(Q^{\prime n})=\delta^n\).

Let \(\varphi\) be a homeomorphic mapping of the ball

\[ B^n\left(\sum_{i=1}^n x_i^2\leq 1\right) \]

onto \(Q^n\), and let \(O'=\varphi(O)\), where \(O\) is the center of \(B^n\). By Lemma 1 one may assume that \(\delta^n\subset \operatorname{int}Q^n\) and \(O'\) is the center of \(\delta^n\). Then \(\varphi^{-1}(\delta^n)\) is an \(n\)-element in \(B^n\), and \(O\subset \operatorname{int}\varphi^{-1}(\delta^n)\). In \(B^n\) choose two \(n\)-dimensional balls with center \(O\):

\[ B'\subset \operatorname{int}\varphi^{-1}(\delta^n);\qquad B''\supset \varphi^{-1}(\delta^n)\cup \varphi^{-1}(K). \tag{2} \]

There exists an isotopy \(h_t:B^n\to B^n\), fixed on \(\partial B^n\), such that \(h_1(B'')=B'\).

\(\theta=h_1^{-1}\varphi^{-1}(\delta^n)\) is an \(n\)-element and, by (2), \(\varphi^{-1}(K)\subset \theta\) and \(B'\subset \theta\). \(F_t=\varphi h_t\varphi^{-1}\) is an isotopy of \(Q^n\) onto itself, fixed on \(\partial Q^n\). Put \(Q^{\prime n}=\varphi(\theta)\); then \(K\subset Q^{\prime n}\). It is easy to verify that \(F_1(Q^{\prime n})=\delta^n\). Setting \(F_t(x)=x\) for \(x\in E^n\setminus Q^n\), we satisfy the conditions of the lemma.

3. Proof of Theorem 1. All \(\overline V_i^{\,k}\) in this case are \(n\)-elements. Let \(\Delta\) be a ball,

\[ \bigcup_{i=1}^{n_1}\overline V_i^{\,1}\subset \Delta. \]

Choose \(n_1\) points on the segment \(E^1\cap \Delta\) and balls \(\delta_i^1\subset \Delta\) with centers at them, such that

\[ \delta_i^1\cap \delta_j^1=\Lambda,\quad i\neq j;\qquad \left(\bigcup_{i=1}^{n_1}\overline V_i^{\,1}\right)\cap \left(\bigcup_{i=1}^{n_1}\delta_i^1\right)=\Lambda. \tag{3} \]

Join in \(\Delta\) each \(\overline V_i^{\,1}\) by a polygonal line \(l_i\) with \(\delta_i^1\) so that \(d_i\cap d_j=\Lambda\), where \(d_i=\overline V_i^{\,1}\cup l_i\cup \delta_i^1\), and for each \(d_i\) choose a neighborhood \(U_i\subset \Delta\) such that

\(\overline U_i\cap \overline U_j=\Lambda,\ i\ne j\). According to Lemma 2, in each \(\overline V_i^1\) one can choose an \(n\)-element \(W_i^1\) so that

\[ \left(\bigcup_j \overline V_j^2\right)\cap V_i^1 \subset \operatorname{int} W_i \subset V_i^1, \tag{4} \]

and construct an isotopy \(F_t^i:\overline U_i\to \overline U_i,\ 0\leqslant t\leqslant 1/2\), fixed on \(\partial U_i\), for which \(F_{1/2}^i(W_i)=\delta_i^1\). Put

\[ F_t(x)=F_t^i(x),\quad x\in \overline U_i,\qquad F_t(x)=x,\quad x\in E^n\setminus \bigcup_{i=1}^n U_i;\quad 0\leqslant t\leqslant 1/2 . \tag{5} \]

Suppose that the isotopy \(F_t\) has been constructed for \(t\leqslant (k-1)/k\); \(\delta_i^r,\ r\leqslant k-1\), are balls with centers on \(E^1\), and

\[ F_t(\overline V_i^r)\subset \delta_j^{\,r-1},\quad \text{if } \overline V_i^r\subset V_j^{r-1},\quad \frac{r-1}{r}\leqslant t\leqslant \frac{k-1}{k}; \]

\[ F_t(x)=F_{(r-1)/r}(x),\quad \text{if } t>\frac{r-1}{r}\ \text{ and }\ x\in E^n\setminus \bigcup_{j=1}^{n_{r-1}} V_j^{r-1} \tag{6} \]

(this is true for \(k=2\)); construct \(F_t\) for \(t\leqslant k/(k+1)\). Choose in each
\(\delta_j^{\,k-1}\setminus F_{(k-1)/k}\left(\bigcup \overline V_i^k\right)\) disjoint balls \(\delta_i^k\) with centers on \(E^1\) for all \(\overline V_i^k\subset V_j^{k-1}\).

Having chosen \(W_i^k\subset F_{(k-1)/k}(V_i^k)\), one on each \(\delta_j^{\,k-1}\), similarly to \(F_t\) for \(0\leqslant t\leqslant 1/2\) in \(\Delta\), constructs an isotopy \(\varphi_t^j:\delta_j^{\,k-1}\to \delta_j^{\,k-1}\), \((k-1)/k\leqslant t\leqslant k/(k+1)\), fixed on \(\partial\delta_j^{\,k-1}\), such that \(\varphi_{k/(k+1)}^j(W_i^k)=\delta_i^k\);
\(\varphi_{k/(k+1)}^jF_{(k-1)/k}(\overline V_s^{\,k+1})\subset \operatorname{int}\delta_i^k\), if \(\overline V_s^{\,k+1}\subset V_i^k\); and put

\[ F_t(x)=\varphi_t^jF_{(k-1)/k}(x),\quad \text{if } x\in F_{(k-1)/k}^{-1}(\delta_j^{\,k-1}); \]

\[ F_t(x)=F_{(k-1)/k}(x),\quad \text{if } x\in E^n\setminus \bigcup_j F_{(k-1)/k}^{-1}(\delta_j^{\,k-1}); \]

\[ \frac{k-1}{k}\leqslant t\leqslant \frac{k}{k+1}. \]

Thus \(F_t\) is defined for \(t<1\), and (6) is satisfied for every \(k\). Put \(F_1(x)=F_{(k-1)/k}(x)\), if \(x\in V_{i_k}^k\setminus \bigcup_j \overline V_j^{\,k+1}\),
\(F_1\left(\bigcap_{k=1}^{\infty}V_{i_k}^k\right)=\bigcap_{k=1}^{\infty}\delta_{i_k}^k,\ k=1,2,\ldots\).

By virtue of (6), \(F_1\) is a homeomorphism for each \(t\), \(F_1(A)=\bigcap_{k=1}^{\infty}\bigcup_{i=1}^{n_k}\delta_i^k\subset E^1\), and \(F_t\) converges uniformly to \(F_1\) as \(t\to 1\); hence \(F_t,\ 0\leqslant t\leqslant 1\), is an isotopy.

4. Proof of Theorem 2. The closed regions \(\overline V_i^k\) in formula (1) may fail to be \(n\)-elements. On the line \(E^1\) place a compactum

\[ C=\bigcap_{k=1}^{\infty}\bigcup_{i=1}^{n_k}\delta_i^k;\quad \delta_i^k\cap \delta_{i'}^k=\Lambda,\quad i\ne i';\quad \delta_i^k\subset \operatorname{int}\delta_j^{\,k-1}, \]

if \(\overline V_i^k\subset V_j^{k-1}\), where all \(\delta_i^k\) are balls with centers on \(E^1\), \(\operatorname{diam}\delta_i^k\to 0\) as \(k\to\infty\), and condition (3) is satisfied. Choose in each \(V_i^1\setminus \bigcup \overline V_j^2\) a ball \(\pi_i^1\). As above, construct \(n_i^1\) regions \(U_i^1\) so that \(\pi_i^1\cup \delta_i^1\subset U_i^1\), \(\overline U_i^1\cap \overline U_j^1=\Lambda,\ i\ne j\), and an isotopy \(\Phi_t,\ 0\leqslant t\leqslant 1/2\), fixed on \(E^n\setminus \bigcup_{i=1}^{n_1}U_i^1\), satisfying the condition

\[ \Phi_t(\overline U_i^1)=\overline U_i^1;\quad \Phi_{1/2}(\delta_i^1)=\pi_i^1. \]

Let \(\pi_j^{r-1}\) be a ball in \(V_j^{r-1}\setminus \bigcup \overline V_i^r\); \(\Phi_t\) has been constructed

for \(t\le (k-1)/k\) and for \(r\le k\):

\[ \begin{aligned} \text{a)}\quad &\Phi_{(r-1)/r}(\delta_i^r)\subset \pi_j^{r-1}\subset V_j^{r-1}\setminus \bigcup \overline V_s^r, \quad \text{if } \delta_i^r\subset \delta_j^{r-1};\\ \text{b)}\quad &\Phi_t(x)=\Phi_{(r-1)/r}(x),\quad \text{if } t>\frac{r-1}{r}\ \text{and}\ \Phi_{(r-1)/r}(x)\in E^n\setminus \bigcup_{j=1}^{n_{r-1}} V_j^{r-1}. \end{aligned} \tag{7} \]

To construct \(\Phi_t\) for \(t\le k/(k+1)\), choose in each
\(V_i^k\setminus \bigcup \overline V_m^{k+1}\) a ball \(\pi_i^k\); then, for all \(V_i^k\), construct domains \(U_i^k\) so that

\[ \begin{gathered} \Phi_{(k-1)/k}(\delta_i^k)\cup \pi_i^k\subset U_i^k;\qquad U_i^k\subset V_j^{k-1},\quad \text{if } V_i^k\subset V_j^{k-1},\\ \overline U_i^k\cap \overline U_{i'}^k=\Lambda,\quad i\ne i'. \end{gathered} \tag{8} \]

By Lemma 2 one can choose in each \(n\)-element \(\Phi_{(k-1)/k}(\delta_i^k)\) an \(n\)-element \(W_i^k\) for which

\[ \Phi_{(k-1)/k}\left[\delta_i^k\cap\left(\bigcup_{m=1}^{n_{k+1}}\delta_m^{k+1}\right)\right]\subset W_i^k\subset \Phi_{(k-1)/k}(\delta_i^k), \tag{9} \]

and construct an isotopy \(\varphi_t,\ (k-1)/k\le t\le k/(k+1)\), such that

\[ \begin{gathered} \varphi_t(\overline U_i^k)=\overline U_i^k,\qquad \varphi_t(x)=x,\quad \text{if } x\in E^n\setminus \bigcup_{i=1}^{n_k} U_i^k,\\ \varphi_{k/(k+1)}(W_i^k)=\pi_i^k. \end{gathered} \tag{10} \]

Put

\[ \Phi_t(x)=\varphi_t\Phi_{(k-1)/k}(x),\qquad \frac{k-1}{k}\le t\le \frac{k}{k+1}. \tag{11} \]

Then (7) is satisfied for \(t\le k/(k+1)\). Note that \(\pi_i^k\) may fail to be contained in \(\pi_j^{k-1}\) when \(V_i^k\subset V_j^{k-1}\).

Thus \(\Phi_t\) is constructed for \(t<1\). From (8) and (10) it follows that if \(\Phi_{(k-1)/k}(x)\in V_j^{k-1}\), then also, for \(t>(k-1)/k\), \(\Phi_t(x)\in V_j^{k-1}\); by virtue of this and (7b), the set of homeomorphisms \(\Phi_t,\ t<1\), converges uniformly as \(t\to 1\) to a continuous mapping \(\Phi_1:E^n\to E^n\), \(\Phi_1(x)=\lim_{t\to1}\Phi_t(x)\). In this case

\[ \begin{gathered} \Phi_1(x)=\Phi_t(x)\big|_{t\ge (k-1)/k}=\Phi_{(k-1)/k}(x), \quad \text{if } x\in \Phi_{(k-1)/k}^{-1}\left[E^n\setminus \bigcup_{i=1}^{n_{k-1}}\overline V_i^{k-1}\right];\\ \Phi_1(x)\in V_j^{k-1},\quad \text{if } x\in \Phi_{(k-1)/k}^{-1}(V_j^{k-1});\\ \Phi_1\left(\bigcap_{k=1}^{\infty}\delta_{i_k}^k\right)=\lim_{k\to\infty}\pi_{i_k}^k= \bigcap_{k=1}^{\infty}V_{i_k}^k,\qquad \delta_{i_k}^k\subset \delta_{i_{k-1}}^{k-1}. \end{gathered} \tag{12} \]

It follows that \(\Phi_1\) is a homeomorphism on the compactum \(C\) and \(\Phi_1(C)=A\). But there may be points \(x\in E^n\setminus C\) such that, for every \(k\), \(\Phi_{(k-1)/k}(x)\in V_{i_k}^k\). By virtue of (8) and (10), \(V_{i_k}^k\subset V_{i_{k-1}}^{k-1}\), and therefore \(\Phi_1(x)\in C\). By virtue of (12), \(\Phi_1\) is a homeomorphism on \(E^n\setminus \Phi_1^{-1}(A)\).

Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR

Received
7 VI 1962

REFERENCES

\(^{1}\) L. Antoine, J. Math. Pures et Appl., Sér. 84, 221 (1921).
\(^{2}\) P. Uryson, Selected Works, 1, 1951, p. 331.
\(^{3}\) R. H. Bing, Pacific J. Math., 11, No. 2, 435 (1961).