2 Smirnov’s compacta, Henderson’s compacta, and essential maps

(A) Smirnov’s compacta $S_{\alpha }$ , $\alpha < \omega _1$ , are defined by transfinite induction (cf. [Reference EngelkingEn2, Example 7.1.33], [Reference PolPo1, Section 2]), $S_1 = I$ , $S_{\alpha + 1} = S_{\alpha } \times I$ and for any limit $\alpha < \omega _1$ , $S_{\alpha }$ is the one-point compactification of the free union $\bigoplus \limits _{\beta < \alpha } S_{\beta }$ .

We shall identify $S_{\alpha }$ with the subspace $S_{\alpha } \times \{ 0 \}$ of $S_{\alpha + 1}$ , and for any limit $\alpha $ and $\beta < \alpha $ , $S_{\beta }$ is embedded in $S_{\alpha }$ as an open subspace.

Open components of $S_{\alpha }$ are closed n-cells, i.e., homeomorphic copies of n-cubes $I^n$ .

In the process of construction of Smirnov’s compacta, we shall also fix, for each open component C of $S_{\alpha }$ , a homeomorphism $e_C:C \to I^n$ , keeping an agreement that, for any open component $D = C \times I$ of $S_{\alpha + 1}$ , C being an open component of $S_{\alpha }$ , $e_D = e_C \times id_I : D \to I^{n+1}$ ( $ id_I$ is the identity on I).

If $\alpha $ is a limit ordinal and C is an open component of $S_{\alpha }$ , C is also an open component of the open copy $S_{\beta }$ in $S_{\alpha }$ containing C, $\beta < \alpha $ , and we let $e_C$ be the map already fixed for this component of $S_{\beta }$ .

For any open component C of $S_{\alpha }$ , $\partial C = e_C^{-1} (\partial I^n)$ is the geometric boundary of C.

Henderson [Reference HendersonHe] (cf. [Reference PolPo1]), extended by transfinite induction each $S_{\alpha }$ to an $AR$ -compactum $H_{\alpha } \supset S_{\alpha }$ such that for each open component C of $S_{\alpha }$ , $C\setminus \partial C$ is open in $H_{\alpha }$ , and distinguished the closed set $\partial H_{\alpha } = H_{\alpha } \setminus \bigcup \{ C\setminus \partial C : C\ \mathrm {is\ an\ open\ component\ of }\ S_{\alpha } \}$ . Let us recall that $H_{\alpha }$ being an $AR$ -compactum means that every continuous map from a closed subset A of a compactum X into $H_{\alpha }$ can be extended from A to a continuous map of X into $H_{\alpha}$ .

(B) Let us recall that a continuous map $f:K \to C$ of a compactum onto a closed cell is essential, if for any continuous $g:K \to C$ coinciding with f on $f^{-1} (\partial C)$ , $g(K) = C$ (cf. [Reference EngelkingEn2] (in [Reference Grispolakis and TymchatynGT, Section 5], essential maps onto closed cells are called $AH$ -essential).

A continuous map $f:K \to I^{\mathbb N}$ of a compactum is essential if for any projection $p_n:I^{\mathbb N} \to I^n$ onto the first n coordinates, $p_n\circ f :K \to I^n$ is essential (cf. [Reference AleksandrovAl, Reference HendersonHe]).

The compacta admitting essential maps onto the Hilbert cube are exactly the strongly infinite-dimensional ones (cf. [Reference AleksandrovAl, Reference EngelkingEn2]).

Henderson [Reference HendersonHe] called a continuous map $f:K\to H_{\alpha }$ of a compactum onto the Henderson compactum essential if for each continuous $g:K \to H_{\alpha }$ coinciding with f on $f^{-1} (\partial H_{\alpha })$ , $g(K) = H_{\alpha }$ (cf. Section 2(A)).

Our reasonings will be based on the following result.

(C) The following theorem of Holsztyński will be an important ingredient of our proof ( Holsztyński calls in [Reference HolsztyńskiHo] essential maps onto cubes “universal” (cf. [Reference Grispolakis and TymchatynGT, Theorem 5.4])).

3 Proof of Theorem 1.1

(A) We adopt the notation from Section 2(A). We shall say that a continuous map $f:K \to S_{\alpha }$ of a compactum onto the Smirnov compactum is determined by products if for every open component C of $S_{\alpha }$ there is a homeomorphism $g:f^{-1} (C) \to T_1 \times T_2 \times \dots \times T_n$ , $n= \mathrm {dim } C$ , onto the product of continua $ T_i$ , and continuous surjections $u_i:T_i \to I$ , $u_i^{-1} (0)$ being a singleton, such that ${f | {f^{-1}(C)} = e_C^{-1} \circ (u_1\times u_2 \times \dots \times u_n ) \circ g}$ .

Let $\alpha = \beta + 1$ , $\beta \geq 1$ , and let us recall that $S_{\beta }$ is identified with $S_{\beta } \times \{ 0 \} \subset S_{\alpha }$ . Then, whenever $f:K \to S_{\alpha }$ is determined by products, so is the restriction $f | f^{-1}(S_{\beta }): f^{-1}(S_{\beta }) \to S_{\beta }$ .

Indeed, if D is an open component of $ S_{\beta }$ , $C= D \times I$ is an open component of $S_{\alpha }$ and in the above notation, g maps $f^{-1} (D)$ onto $T_1 \times T_2 \times \dots \times T_{n-1} \times \{ a\}$ , $ \{ a\}= u_n^{-1}(0)$ .

Clearly, if $f:K \to S_{\alpha }$ is determined by products, $\alpha $ is limit, $\beta < \alpha $ , and $S_{\beta }$ is the open copy of the Smirnov compactum in $S_{\alpha }$ , then $f | f^{-1}(S_{\beta }): f^{-1}(S_{\beta }) \to S_{\beta }$ is also determined by products.

This implies readily by transfinite induction on $\alpha $ that if a compactum K admits a continuous map onto $S_{\alpha }$ determined by products and $\beta < \alpha $ , then there is a compactum L in K admitting a continuous map onto $S_{\beta }$ determined by products.

Finally, by the Holsztyński Theorem 2.3 and Remark 2.1, if $f:K \to H_{\alpha }$ is a continuous map of a compactum onto the Henderson compactum and for some compactum $L \subset K$ , the restriction $f | L : L \to S_{\alpha }$ is determined by products, then f is an essential map onto $H_{\alpha }$ .

(B) Let X, Y be compacta of positive dimension and let $h: X \times Y \to X$ be a light map. We shall fix metrics $d_X$ , $d_Y$ generating the topology of X and Y, respectively, and we endow $ X \times Y $ with the metric

$d_{ X \times Y } \big ( (x_1,y_1), (x_2,y_2) \big ) = \mathrm {max} \big ( d_X(x_1,x_2), d_Y (y_1,y_2) \big )$ .

We shall use the uniform continuity of h with respect to the metric $d_{ X \times Y }$ in $ X \times Y$ and the metric $d_ X$ in X.

We shall check by transfinite induction the following claim.

The claim is true for $\alpha = 1$ . Indeed, given $\varepsilon> 0$ , we pick a nontrivial continuum $K \subset X$ with $\mathrm {diam} K < \varepsilon $ and f is the identity on K (recall that $S_1 = I$ ).

Assume that Claim 3.1 is true for all $\beta < \alpha $ , $\alpha $ being a countable ordinal.

Assume first that $\alpha = \beta + 1$ . Let us fix $\varepsilon> 0$ and let $\delta> 0$ be such that h takes sets in $X \times Y$ of diameter $< \delta $ to sets in X of diameter $< \varepsilon $ . By the inductive assumption, there is a compactum L, a continuous surjection $u:L \to S_{\beta }$ determined by products and a light map $g:L \to X$ with $\mathrm {diam}\big ( g(L) \big )< \delta $ . Let T be a continuum in Y with $\mathrm {diam}T < \delta $ and let $w:T \to I$ be a continuous surjection such that $w^{-1} (0)$ is a singleton.

Then, the continuous map $u \times w:L \times T \to S_{\beta } \times I = S_{\beta + 1 } = S_{\alpha }$ is determined by products and $\mathrm {diam} \big ( h\big ( g(L) \times T \big ) \big ) < \varepsilon $ . The continuous map $f=h\circ (g\times id_T) : L \times T \to X$ is light ( $ id_T$ is the identity on T), as the composition of two light maps and $\mathrm {diam} f(L \times T) < \varepsilon $ .

Assume now that $\alpha $ is a limit ordinal. Let $\varepsilon> 0$ and $\delta> 0$ be as in the previous case. Let us list all ordinals less than $\alpha $ as $\alpha _1, \alpha _2, \dots $ and let us choose ordinals $\beta _1 < \beta _2 < \dots $ with $\alpha _n \leq \beta _n < \alpha $ .

The inductive assumption provides compacta $L_n$ which have continuous maps onto $ S_{\beta _n }$ determined by products, and light maps $f_n:L_n \to X$ with $\mathrm {diam} f_n(L_n) < {1\over n}$ .

By compactness of X, passing if necessary to a subsequence, we can ensure that there is $a\in X$ such that ${\delta \over 2}> \mathrm {diam} \big ( \{ a \} \cup f_n(L_n) \big ) \to 0$ .

The observation made in Section 3(A), allows one to pick for each n a compactum $M_n$ in $L_{n }$ which has a continuous map onto $ S_{\alpha _n }$ determined by products.

The one-point compactification K of the free union $\bigoplus \limits _n M_n$ has a continuous map onto $S_{\alpha }$ determined by products, $S_{\alpha }$ being the one-point compactification of the free union $\bigoplus \limits _n S_{\alpha _n }$ (the map takes $M_n$ onto $S_{\alpha _n }$ and the point at infinity of K to the point at infinity of $S_{\alpha }$ ).

Let $b_n \to b$ be a sequence in Y with all $b, b_1, b_2, \dots $ distinct and $\mathrm {diam} \{ b, b_1, b_2, \dots \} < \delta $ . Then, $C = \{ (a,b) \} \cup \bigcup \limits _{n=1}^\infty \big ( f_n(M_n) \times \{ b_n \} \big ) $ is topologically the one-point compactification of $\bigoplus \limits _n f_n(M_n)$ , $(a,b)$ being the point at infinity, hence the map $u:K \to C$ taking the point at infinity of K to $(a,b)$ and such that $u(x) = \big ( f_n (x), b_n),$ whenever $x \in M_n$ , is a light map. The composition $h \circ u : K \to X$ is light and $\mathrm {diam} \big ( h \circ u (K) \big ) < \varepsilon $ .

(C) Having checked Claim 3.1, we obtain for each $\alpha < \omega _1$ a compactum ${K_{\alpha } \subset I^{\mathbb N}}$ which has a continuous map onto $S_{\alpha }$ determined by products and a light map $f_{\alpha } : K_{\alpha } \to X$ (the additional condition involving $\varepsilon> 0$ was only needed to carry out the inductive construction).

We shall verify the following claim.

To that end, let us take a continuous surjection $\Phi : 2^{\mathbb N} \to \mathcal {K} (I^{\mathbb N} \times X)$ of the Cantor set onto the space of non-empty compact sets in $I^{\mathbb N} \times X$ , equipped with the Vietoris topology (cf. [Reference EngelkingEn1, Problem 2.7.20, Problem 4.5.23]), and let

$H = \{ (t,x,y): (x,y) \in \Phi (t) \}$ , $p(t,x,y) = y$ .

Then, $H= \bigcup \limits _{t \in 2^{\mathbb N} } \{ t \} \times \Phi (t) \subset 2^{\mathbb N} \times I^{\mathbb N} \times X $ is a compactum and $p:H \to X$ is a continuous map whose restriction to $ \{ t \} \times \Phi (t)$ is the projection of $\Phi (t)$ onto the second coordinate.

The graph $\mathrm {Gr} (f_{\alpha } ) = \{ \big ( x, f_{\alpha } (x) \big ) : x \in K_{\alpha } \}$ is a compact set in $ I^{\mathbb N} \times X $ , hence $\mathrm {Gr} (f_{\alpha }) = \Phi (t_{\alpha } ) $ for some $t_{\alpha } \in 2^{\mathbb N}$ . The compactum $ \{ t_{\alpha } \} \times \mathrm {Gr} (f_{\alpha })$ is homeomorphic to $K_{\alpha } $ and the restriction of p to this compactum is light, as $p\big ( t_{\alpha } , x, f_{\alpha } (x) \big ) = f_{\alpha } (x)$ .

Let $u:H\to Z$ , $w:Z \to X$ be the monotone-light factorization of p. i.e., the fibers of u are connected and the fibers of w are zero-dimensional (cf. [Reference EngelkingEn1, Theorem 6.2.23]). In particular, the map u is obtained by matching each nontrivial continuum in every fiber $p^{-1}(x)$ to a point.

Now, any continuum in H is contained in some $ \{ t \} \times \Phi (t)$ , and if a continuum is contained in $ \{ t_{\alpha } \} \times \mathrm {Gr} (f_{\alpha })$ and taken by p to a point, then the continuum is a singleton, the restriction of p to $ \{ t_{\alpha } \} \times \mathrm {Gr} (f_{\alpha })$ being light. It follows that u is injective on $ \{ t_{\alpha } \} \times \mathrm {Gr} (f_{\alpha })$ and in effect, $u \big ( \{ t_{\alpha } \} \times \mathrm {Gr} (f_{\alpha }) \big )$ is a copy of $K_{\alpha }$ in Z.

(D) With Claim 3.2 in hand, we are ready to complete the proof of Theorem 1.1. Let $w:Z\to X$ be as in Claim 3.2. For each $\alpha < \omega _1$ we have a compactum $L_{\alpha } \subset Z$ and a continuous map $g_{\alpha } : L_{\alpha } \to S_{\alpha }$ onto the Smirnov compactum determined by products.

Let $Z^* = Z \bigoplus 2^{\mathbb N} $ and let $v_{\alpha } :2^{\mathbb N} \to \partial H_{\alpha }$ be a continuous surjection, $\alpha < \omega _1$ . Since $H_{\alpha }$ is an $AR$ , the combination of $g_{\alpha }$ and $v_{\alpha }$ can be extended from $L_{\alpha } \cup 2^{\mathbb N}$ to a continuous surjection $\tilde {g}_{\alpha } : Z^* \to H_{\alpha }$ . By the observation at the end of Section 3(A), $\tilde {g}_{\alpha }$ is essential. By Theorem 2.2, $Z^*$ is strongly infinite-dimensional, and so is Z.

By [Reference PolPo2, Corollary 2.3], there is a countable-dimensional $A\subset X$ such that $X\setminus A$ contains no one-dimensional subsets. The map w being light, $w^{-1}(A)$ is countable-dimensional and therefore, by [Reference EngelkingEn2, Proposition 6.1.17], there is a nontrivial continuum $C\subset Z \setminus w^{-1}(A)$ . The nontrivial continuum $w(C)$ contains no one-dimensional subsets.

4 Comments

The proofs of the results stated in this section will be presented by the second author in a separate paper [Reference ReńskaRe].

4.1. Embedding of countable products If $h:X \times Y \to X$ in Theorem 1.1 is an embedding and for some nontrivial continuum K, Y contains copies of K with arbitrarily small diameters, then $K^{\mathbb N}$ embeds in X.

4.2. On the transfinite dimension ind Let $h:X \times Y \to X$ in Theorem 1.1 be an embedding. Combining the approach of this article with a subtle theorem of Levin [Reference LevinLe], one can prove that X contains countable-dimensional compacta with arbitrarily large transfinite dimension ind (cf. [Reference EngelkingEn2]).

4.3. On non-compact spaces Let X be a completely metrizable separable space and let $h:G\to X$ be a continuous injective map of an open set G in $X \times X$ such that the image $h(G)$ is dense in X.

If K is a nontrivial continuum such that each non-empty open set in X contains a topological copy of K, then $K^{\mathbb N}$ embeds in X.