Reports of the Academy of Sciences of the USSR

1. Volume 114, No. 3

MATHEMATICS

LYUDMILA KELDYSH

TRANSFORMATION OF MONOTONE IRREDUCIBLE MAPPINGS INTO MONOTONE-OPEN ONES AND A MONOTONE-OPEN MAPPING OF A CUBE ONTO A CUBE OF HIGHER DIMENSION

(Presented by Academician P. S. Aleksandrov on 21 III 1957)

In the present paper we indicate a method by which, under certain conditions, one can obtain from a monotone irreducible mapping of a continuum (X) a monotone-open mapping of (X). Since in many cases the problem of constructing a monotone irreducible mapping is simpler than the problem of constructing a monotone-open mapping, Theorems 1 and 2 give a method for constructing various examples of monotone-open mappings, in particular, monotone-open mappings that raise dimension. From Theorem 1 there follows a positive solution of P. S. Aleksandrov’s problem on the existence of an open mapping of a (p)-dimensional cube onto a (q)-dimensional one for (q>p\geq 3). We note that a solution of this problem was announced by R. D. Anderson in 1953 ((^{1})) and again in 1956 ((^{2})), without indication of a method of construction, but the exposition of the proof has not yet appeared in print.

Here we shall describe briefly the idea of the proof of Theorems 1 and 2; a detailed exposition will be printed in the journal Matematicheskii Sbornik.

Recall that for an irreducible mapping (f) of a compactum (X) onto a compactum (Y), in (X) there is everywhere dense the set (E^{-1}) of points of uniqueness, i.e. such points that (x=f^{-1}f(x)); on (E^{-1}) the mapping (f) is a homeomorphism, and (E^{-1}), as also (E=f(E^{-1})), is a set of type (G_\delta).

Theorem 1. Let (f) be a monotone irreducible mapping of a continuum (X) onto a manifold (with boundary or without boundary) (Y) of dimension (n\geq 3), and suppose that the intersection of every domain in (Y) with the set (E) of points of uniqueness of (f) is connected. Then, whatever the number (\varepsilon>0), there exists a continuous (\varepsilon)-shift (\Phi) of the manifold (Y) onto itself such that the superposition (F=\Phi f) is a monotone-open mapping of (X) onto (Y).

Theorem 2. Let (f\colon X\to Y) be a monotone irreducible mapping of a continuum (X) onto a locally connected continuum (Y) such that no connected open set (domain) (U) in (Y) is separated by a simple arc. If the intersection (E\cap U) of any connected open set with the set of points of uniqueness is connected, then there exists a monotone mapping (\Phi) of the continuum (Y) onto a continuum (Z) such that the superposition (F=\Phi f) is a monotone-open mapping of the continuum (X) onto the continuum (Z), and
[
\dim Z\geq \dim Y-1.
]

We denote by (h(A,B)) the deviation of the sets (A) and (B), i.e. the upper bound of the distances from points of one of them to the other. By (X_f) we denote the continuous decomposition of the compactum (X) corresponding to the continuous mapping (f).

The proof of Theorems 1 and 2 is based on the following two lemmas.

Lemma 1. Let (f) be a monotone irreducible mapping of a compactum (X) onto a locally connected compactum (Y), no domain of which (U) is separated by a simple arc, and suppose that the intersection (E\cap U) of the set of points of uniqueness—

is connected with any domain. Let
[
Y=\bigcup_{i=1}^{N}\Delta_i
]
be an (\varepsilon)-covering of multiplicity (m) by locally connected closures of domains, and suppose that (f^{-1}(\Delta_i)) is contained in a neighborhood of diameter (<\varepsilon) of an element of the decomposition (\xi_i\in X^f). Then there exists a continuous decomposition (Y^\varphi), having only a finite number of nondegenerate elements, each of which is the sum of not more than (m) simple arcs, contained in (E) and having one common end, and a covering
[
Y=\bigcup \delta_r
]
of multiplicity (m), where (\delta_r) contains only one point (f(\xi_i)), (i=1,\ldots,N), or a nondegenerate element (\eta_r\in Y^\varphi); (f^{-1}(\delta_r)) is a connected neighborhood of diameter (<\varepsilon) for (\xi_i=\xi_i) or for (\xi_r=f^{-1}(\eta_r)); from (\delta_r\cap\delta_{r'}\ne\Lambda) it follows that (h(\xi'r,\xi')<\varepsilon), and if
[
\bigcap_{k=1}^{p}\delta_{r_k}\ne\Lambda
\quad\text{and}\quad
\bigcup_{k=1}^{p}\delta_{r_k}
]
intersects (\Delta_{i_\nu}), (\nu=1,\ldots,q), then
[
\bigcap_{\nu=1}^{q}\Delta_{i_\nu}\ne\Lambda.
]

Lemma 2. Let (f) be a monotone irreducible mapping of a compactum (X) onto an (n)-dimensional locally connected compactum (Y), no domain of which (U) is split by a simple arc, and let the intersection (U\cap E) be connected. Let
[
Y=\bigcup_i\Delta_i
]
be a closed covering of multiplicity (m\le n+1), and suppose that (f^{-1}(\Delta_i)) is a connected neighborhood of diameter (<\varepsilon) for an element of the decomposition (\xi_i\in X^f), and that from (\Delta_i\cap\Delta_j\ne\Lambda) it follows that (h(\xi_i,\xi_j)<\varepsilon). Then, whatever number (\varepsilon'>0) may be, there exists a continuous decomposition (Y^\psi), having only a finite number of nondegenerate elements (\eta_r), each of which is the sum of not more than (m) simple arcs having a common end (y_r) and contained in (E) except for the point (y_r), and a covering
[
Y=\bigcup \delta_r
]
of multiplicity (\le n+1), where (\delta_r) are locally connected closures of domains, each (\delta_r) contains only one point (f(\xi_i)) or an element (\eta_r); (f^{-1}(\delta_r)) is a neighborhood of diameter (<\varepsilon') for (\xi'r=\xi_i) or (\xi'_r=f^{-1}(\eta_r)); from (\Delta_i\cap\delta_r\ne\Lambda) it follows that (h(\xi_i,\xi'_r)<2\varepsilon), and if
[
\bigcap\ne\Lambda}^{p}\delta_{r_k
\quad\text{and}\quad
\bigcup_{k=1}^{p}\delta_{r_k}
]
intersects (\Delta_{i_\nu}), (\nu=1,\ldots,q), then
[
\bigcap_{\nu=1}^{q}\Delta_{i_\nu}\ne\Lambda.
]

We indicate quite briefly the idea of the proof of Lemma 1. It is carried out by induction on the multiplicity (m) of the covering. If (m=1), then one may put (\delta_i=\Delta_i), and all elements of the decomposition (Y^\varphi) are points. Let (m>1). We enclose each intersection (\Delta_{i_1}\cap\cdots\cap\Delta_{i_m}) in an open set whose closure (\pi_{i_1\ldots i_m}) is locally connected and which is contained inside
[
\bigcup_{\nu=1}^{m}\Delta_{i_\nu},
]
and moreover so that distinct (\pi_{i_1\ldots i_m}) do not intersect one another. Each (\pi_{i_1\ldots i_m}) is the sum of a finite number of components (\pi_\mu). In each (\pi_\mu), for each (\pi_{i_1\ldots i_m}), we choose an open set (\sigma_\mu) so that
[
\Delta_{i_1}\cap\cdots\cap\Delta_{i_m}\cap\pi_\mu\subset\sigma_\mu
]
and
[
Y'=Y\setminus\bigcup\sigma_\mu
]
is locally connected;
[
Y'=\bigcup_{i=1}^{N}(\Delta_i\cap Y')
]
has multiplicity (<m), and for (Y') there exists a covering
[
Y'=\bigcup\delta_r
]
satisfying the conditions of the lemma, and a mapping (\varphi'). All nondegenerate elements of the decomposition (Y'^{\varphi'}) will also be elements of the decomposition (Y^\varphi).

The subsequent construction is made for each (\pi_\mu); therefore in what follows we omit the index (\mu). In each (\delta_r\cap\pi) we choose a point of uniqueness (y_r) and draw in (\delta_r) a tube (\tau_r), which is a neighborhood of diameter (<\varepsilon'<\varepsilon/2) of some simple arc (l_r\subset E), one end of which is (y_r), while the other lies in (\delta_r), and such that, if (\eta_r) is an element of the decomposition (Y'^{\varphi'}) whose neighborhood is (\delta_r), then (f^{-1}(\eta_r)) is contained in the (\varepsilon')-neighborhood of (f^{-1}(\tau_r)) and (\tau_r\cap\eta_r=\Lambda);
[
\tau_r=\bigcup_k u_{rk},
]
where (u_{rk}) is a neighborhood of diameter (<\varepsilon') of some point (y\in l_r) and (\operatorname{diam} f^{-1}(u_{rk})<\varepsilon'), with (u_{rk}\cap u_{rk'}=\Lambda) if (|k-k'|>1). The various tubes (\tau_r) for all possible (\pi_\mu) and (\delta_r) may be chosen so that they do not intersect and do not split (\pi_\mu) and (\delta_r). Then we put:
[
\theta=\pi\cup\bigcup_{\tau_r\cap\pi\ne\Lambda}\tau_r.
]

By construction,

[
\theta_\mu \cap \theta_{\mu'} = \Lambda .
]

From some point (y\in \pi\cap E) we draw (m) simple arcs

[
l_\nu \subset (\pi\cup \tau_{i_\nu})\cap E,
]

where (\tau_{i_\nu}\subset \delta_{i_\nu}) and (\delta_{i_\nu}\supset f(\xi_{i_\nu})). Let

[
L=\bigcup_{\nu=1}^{m} l_\nu ;
]

this is an element of the decomposition (Y^\varphi). To construct the remaining nondegenerate elements of (Y^\varphi) contained in (\theta), we choose in

[
\pi\cup \bigcup_{\nu=1}^{m}\tau_{i_\nu}
]

a tube

[
\tau=\bigcup_{k=1}^{k_0} u_k,
]

which satisfies the same conditions as the tubes (\tau_r); that is, a neighborhood of diameter (<\varepsilon') of a simple arc (l\subset E), one end of which, (x\in u_1\subset \pi), and moreover

[
h(L,\tau)<\varepsilon',\qquad L\cap \tau=\Lambda .
]

Each nondegenerate element (\eta_\rho\in \theta^\varphi), (\eta_\rho\ne L), is a simple arc, and for some (r) and (p)

[
\eta_\rho\subset \tau_r\cup \tau \cup \bigcup_{k=1}^{p}u_k,\qquad p\le k_0,\qquad \tau_r\cap \pi\ne \Lambda .
]

Here

[
\eta_\rho\subset E,
]

(\eta_\rho) meets all (u_{rj}), one end of (\eta_\rho) lies in (\tau_r\setminus \pi), and the other in (u_p). The elements of the cover (\delta_\rho\subset \theta) are constructed so that (\delta_\rho) and (\delta_{\rho'}) may intersect only if the ends of (\eta_\rho) and (\eta_{\rho'}) lying in (u_k) and (u_{k'}) are contained in summands (u_k) and (u_{k'}) with (|k-k'|\le 1). For the construction, first the case is considered when the multiplicity of

[
\bigcup_r(\theta\cap \partial_r)
]

is equal to 1, and then the general case.

The construction and the proof are cumbersome and are carried out with the aid of additional inductive assumptions of a technical nature.

For the proof of Lemma 2 we choose the number (\varepsilon') so that it is less than half the Lebesgue number for the given cover

[
Y=\bigcup \Delta_i,
]

and we choose for (Y) an (\varepsilon')-cover

[
\bigcup \Delta'_r
]

of multiplicity (\le n+1) so that (f^{-1}(\Delta'r)) is a neighborhood of diameter (<\varepsilon') of some element (\xi_r\in X^f). Each nondegenerate element (\eta_r\in Y^\psi) is a sum of simple arcs (l) having a common end

[
y_r=\bar f(\xi_r),
]

with

[
l_{i_\nu}\setminus y_r \subset E\cap(\Delta_{i_\nu}\cup \Delta'_r),
]

if

[
\Delta_{i_\nu}\cap \Delta'_r\ne \Lambda
]

and (\xi_{i_\nu}) is contained in the (\varepsilon')-neighborhood (f^{-1}(l_{i_\nu}));

[
\delta_r\subset \Delta'r\cup \bigcup\nu \tau_{r i_\nu},
]

where the tubes (\tau_{r i_\nu}) are constructed in the same way as for Lemma 1. The construction is performed in each (\Delta_i) first for (\Delta'r) intersecting (\Delta_i\supset f(\xi_i)), and then, successively, for (\Delta'_r) intersecting some (\Delta\rho) for which the construction has been performed.

For the proof of Theorem 1 it is easy to show that if (Y) is a manifold, then a simple arc lying in (E) can be constructed so that it has arbitrarily small neighborhoods homeomorphic to a ball. All tubes (\tau) in Lemmas 1 and 2 are likewise homeomorphic to a ball. We show that in this case the mappings (\varphi) and (\psi) can be realized as (\varepsilon)-shifts of (Y) onto itself (for (\varphi), for example, this is done by induction on (m), on the basis of the fact that a small neighborhood (\theta) can be mapped homeomorphically onto itself so that (\theta) passes into (\pi)).

We choose a number (\varepsilon) and a convergent series

[
\varepsilon=\sum_{k=0}^{\infty}\varepsilon_k .
]

Having chosen an arbitrary (\varepsilon_0)-cover of (Y) and applying Lemmas 1 and 2 successively, we construct a sequence of continuous mappings:

[
f_{2k-1}=\varphi_k\psi_{k-1}\varphi_{k-1}\cdots \psi_1\varphi_1 f;
]

[
f_{2k}=\psi_k\varphi_k\psi_{k-1}\cdots \psi_1\varphi_1 f.
]

The mapping (f_{2k+1}) and the cover

[
Y=\bigcup_j \Delta_{2k+1,j}
]

are constructed by applying Lemma 1 to the mapping (f_{2k}) and the cover

[
Y=\bigcup_i \psi_k(\Delta_{2k,i}),
]

where

(\operatorname{diam}\Delta_{2k,i}<2\varepsilon_{k-1}), (\operatorname{diam}\psi_k(\Delta_{2k,i})<\varepsilon_k). The mapping (f_{2(k+1)}) is constructed by applying Lemma 2 to the mapping (f_{2k+1}) and the covering (Y=\bigcup \varphi_{k+1}(\Delta_{2k+1,j})), with (\operatorname{diam}\Delta_{2k+1,j}<2\varepsilon_k) and (\operatorname{diam}\varphi_{k+1}(\Delta_{2k+1,j})<\varepsilon_k). The mappings (\varphi_k) and (\psi_k) are (\varepsilon_k)-shifts of (Y) onto itself. Therefore the sequence of mappings (f_m) converges uniformly to a continuous mapping (F) of the continuum (X) onto the manifold (Y). Putting

[
\Phi=\lim_{k\to\infty}\varphi_k\psi_k\cdots\varphi_1\psi_1,
]

we obtain

[
F=\Phi f .
]

We show that the mapping (F) is monotone-open.

Corollary 1. There exists a monotone-open mapping of the (p)-dimensional cube (C_p) onto the (q)-dimensional cube (C_q), (q>p\geqslant 3).

It suffices to consider the case (p=3,\ q=4), since the general case is reduced to it. In ((^3,^4)) we constructed a monotone irreducible mapping (f) of the three-dimensional cube (C_3) onto the four-dimensional cube (C_4). It is easy to show that, for the mapping (f), the set of multiple points (C_4\setminus E) (of type (F_\sigma)) has dimension 2; consequently, (E) is connected in every domain, and the conditions of Theorem 1 are satisfied; hence there exists an (\varepsilon)-shift (\Phi) of the cube (C_4) onto itself such that (\Phi f) is a monotone-open mapping of the cube (C_3) onto the cube (C_4).

For the proof of Theorem 2 we construct a continuous decomposition (Y^\Phi), again successively applying Lemmas 1 and 2. It is easy to show that all elements of the continuous decomposition (Y^\Phi) are one-dimensional continua. Consequently,

[
\dim \Phi(Y)\geqslant \dim Y-1 .
]

Theorem 2 is also true in the case when (n=\infty); in this case the multiplicities of the coverings increase to infinity.

Applying Theorem 2 to the example constructed by us of a monotone irreducible mapping of a cube onto a continuum, in a neighborhood of every point of which there is contained a topological Hilbert parallelepiped (5), we find:

Corollary 2. There exists a monotone-open mapping of the three-dimensional cube onto a continuum, in every neighborhood of each point of which there is contained a topological Hilbert parallelepiped.

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

Received
18 II 1957

CITED LITERATURE

1. R. D. Anderson, Bull. Am. Math. Soc., 59, 3, 243 (1953).
2. R. D. Anderson, Proc. Nat. Acad. Sci. USA, 42, 6, 347 (1956).
3. L. V. Keldysh, Matem. sborn., 41 (83), 2, 129 (1957).
4. L. V. Keldysh, DAN, 103, No. 6, 957 (1955).
5. L. V. Keldysh, Tr. Matem. inst. im. V. A. Steklova, 38, 72 (1951).