Locally $n$-connected compacta and $UV^n$-maps

We provide a machinery for transferring some properties of metrizable $ANR$-spaces to metrizable $LC^n$-spaces. As a result, we show that for complete metrizable spaces the properties $ALC^n$, $LC^n$ and $WLC^n$ coincide to each other. We also provide the following spectral characterizations of $ALC^n$ and cell-like compacta: A compactum $X$ is $ALC^n$ if and only if $X$ is the limit space of a $\sigma$-complete inverse system $S=\{X_\alpha, p^{\beta}_\alpha, \alpha<\beta<\tau\}$ consisting of compact metrizable $LC^n$-spaces $X_\alpha$ such that all bonding projections $p^{\beta}_\alpha$, as a well all limit projections $p_\alpha$, are $UV^n$-maps. A compactum $X$ is a cell-like (resp., $UV^n$) space if and only if $X$ is the limit space of a $\sigma$-complete inverse system consisting of cell-like (resp., $UV^n$) metric compacta.


Introduction
Following [7], we say that a space M is weakly locally n-connected (briefly, W LC n ) in a space Y if M ⊂ Y is closed and for every point x ∈ M and its open neighborhood U in M there exists a neighborhood V of x in M such that any map from S k , k ≤ n, into V is null-homotopic in U , where U is any open in Y set with U ∩ M = U (in such a case we say that U is an open extension of U in Y ). Dranishnikov [5] also suggested the following notion: a space M is approximately locally nconnected (briefly, ALC n ) in a space Y if M ⊂ Y is closed and for every point x ∈ M and its open neighborhood U in M there exists a neighborhood V of x in M such that for any open in Y extension U of U there exists and open in Y extension V of V with any map from S k , k ≤ n, into V being null-homotopic in U. One can show that if M is metrizable (resp., compact), then M is W LC n in a given space Y , where Y is a metrizable (resp., compact) ANR, if and only if M is W LC n in any metrizable (resp., compact) ANR containing M as a closed set. The same is true for the property ALC n . So, the definitions of W LC n and ALC n don't depend on the ANR-space containing M, and we say that M is W LC n (resp., ALC n ).
Dranishnikov [7] proved that both properties W LC n and LC n are identical in the class of metrizable compacta. It also follows from Gutev [10] and  that this remains true for complete metrizable spaces. One of the main result in this paper, Theorem 2.7, shows that all properties W LC n , LC n and ALC n coincide for completely metrizable spaces. The proof of Theorem 2.7 is based on the technique, developed in Section 2, for transferring properties of metrizable ANR's to LC n -subspaces (in this way well known properties of metrizable LC n -spaces can be obtained from the corresponding properties of metrizable ANR's, see for example Proposition 2.2). Section 2 contains also a characterization of metrizable LC n -spaces whose analogue for ANR's was established by Nhu [12].
It is well known that the class of metrizable LC n -spaces are exactly absolute neighborhood extensors for (n + 1)-dimensional paracompact spaces, and this is not valid for non-metrizable spaces. Outside the class of metrizable spaces we have the following characterization of ALC n compacta (Theorem 3.1): A compactum X is ALC n if and only if X is the limit space of a σ-complete inverse system S = {X α , p β α , α < β < τ } consisting of compact metrizable LC n -spaces X α such that all bonding projections p β α , as a well all limit projections p α , are UV nmaps. A similar spectral characterization is obtained for cell-like or UV n compacta, see Theorem 3.3. Both Theorem 3.1 and Theorem 3.3 provide different classes of compacta C and corresponding classes of maps M adequate with C in the following sense (see Shchepin [18]): A compactum X belongs to C if and only if X is the limit space of a σ-complete inverse system S = {X α , p β α , α < β < τ } consisting of compact metrizable X α ∈ C with all bonding projections p β α being from M. For example, according to Theorem 3.1, the class of ALC n compacta is adequate with the class of UV n -maps.
Recall that a closed subset A ⊂ X is UV n in X (resp., cell-like in X) if every neighborhood U of A in X contains a neighborhood V of A such that, for each 0 ≤ k ≤ n, any map f : S k → V is null-homotopic in U (resp., A is contractible in every neighborhood of A in X). A space X is said to be UV n (resp., cell-like) provided it is UV n (resp., cell-like) in some ANR-space containing X as a closed set. It is well known that for metrizable or compact X this definition does not depend on the ANR-spaces containing X as a closed set, see for example [17]. A map f : X → Y between compact spaces is called UV n (resp., cell-like) if all fibres of f are UV n (resp., cell-like).
Theorem 3.1 yields that any compact LC n -space is ALC n . It is interesting to find an example of an ALC n -compactum which is not LC n (obviously, such a compactum should be non-metrizable).

Metrizable ALC n -spaces
We are going to establish some properties of metric LC n -spaces using the corresponding properties of metric ANR-spaces. Recall that a map f : X → Y is n-invertible if for every Tychonoff space Z with dim Z ≤ n and any map g : The next theorem follows from a stronger result due to Pasynkov [15,Theorem 6], and its proof is based on Dranishnikov results [6, Theorem 1] and [7, Theorem 1.2] (such a result concerning the extension dimension with respect to quasi-finite complexes was established in [14,Proposition 2.7]). We provide here a proof based on factorization theorems.
Theorem 2.1. Every Tychonoff space M is the image of a Tychonoff space X with dim X ≤ n under a perfect n-invertible map. In case M is metrizable, X can be supposed to be also a metrizable space with w(X) = w(M).
Proof. Let M be a Tychonoff space of weight τ . Consider all couples (Z α , f α ), where Z α is a Tychonoff space of weight w(Z α ) ≤ w(βM), dim Z α ≤ n and f α is a map from Z α into M (here βM is theČech-Stone compactification of M). Denote by Z the disjoint sum of all spaces Z α . Obviously, there is a natural map f : Z → M such that f |Z α = f α for all α. Let f : βZ → βM be the extension of f . Then, by the Mardešić's factorization theorem [11], there exists a compactum X of weight w( X) ≤ w(βM) with dim X ≤ dim βZ = n and maps h : βZ → X, g : X → βM such that g • h = f . Let X = g −1 (M) and g = g|X. According to Corollary 6 and Main Theorem from [15], dim X ≤ n. To show that g is n-invertible, suppose f 0 : Z 0 → M is a map with dim Z 0 ≤ n. Applying again the Mardešić's factorization theorem for the map f 0 : βZ 0 → βM, we obtain a compactum K and maps h 1 : βZ 0 → K and f 2 : K → βM such that dim K ≤ n, w(K) ≤ w(βM) and f 2 • h 1 = f 0 . Then, as above, Z ′ = f −1 2 (M) is a space of dimension ≤ n and weight ≤ w(βM). So, there exists α * and a homeomorphism j : If M is metrizable, the proof is simpler. Indeed, in this case P = f −1 (M) is a space of dimension ≤ n and the restriction f |P is a perfect map. So, by Pasynkov's factorization theorem [16], there exists a metrizable space X and maps h : P → X, g : X → M such that g • h = f , w(X) ≤ w(M) and dim X ≤ n. Then g is a perfect map because so is f |P , and according to the above arguments, g is n-invertible.
The next proposition shows that Theorem 2.1 allows some properties of metrizable ANR-spaces to be transferred to metrizable LC n -spaces. Proof. We embed M as a closed subset of a metrizable ANR-space P and let p : Y P → P be a perfect (n + 1)-invertible surjection such that Y P is a metrizable space of dimension ≤ n + 1 (see Theorem 2.1). Since We also need the following property of metrizable LC n -spaces. Proof. Let p : Y P → P be a perfect (n + 1)-invertible surjection such that Y P is a metrizable space of dimension ≤ n + 1, and q : It is easily seen that the maps h and h 1 are β-near. So, h and h 1 are homotopic in W , and hence in P .
Corollary 2.4. If M and P are metrizable LC n -spaces such that M ⊂ P is closed, then M is ALC n in P .
Proof. Since M is LC n , for every x ∈ M and its open neighborhood U in M there exists a neighborhood V of x in M such that any map of a k-sphere, k ≤ n, into V is contractible in U. Let U ⊂ P and G ⊂ U be open in P extensions of U and V , respectively. Then both V and G are LC n (as open subsets of LC n -spaces), and V is closed in G. So, there is an open extension V ⊂ G satisfying the conclusion from Proposition 2.4. Consequently, any map g : S k → V , 0 ≤ k ≤ n, is homotopic in G to a map g 1 : S k → V . Since g 1 is homotopic in U to a constant map, we obtain that g is homotopic in U to a constant map. Proposition 2.5. Let P be metrizable and M ⊂ P be a closed and LC n -set. Then every closed set Proof. Let p : Y P → P be a perfect (n + 1)-invertible surjection such that Y P is a metrizable space of dimension ≤ n + 1, and q : Then extend g to a map g 1 : B k+1 → U and take a lifting Next lemma is a non-compact analogue of Lemma 2.1 from [7]. Lemma 2.6. Suppose both M and P are metric LC n−1 -spaces such that M is a closed subset of P . Then for every ǫ > 0 there exists a neighborhood U ǫ (M) in P such that for any map ϕ : Proof. The proof is similar to that one of Proposition 2.5, the only difference is that the map p : Y P → P is n-invertible and dim Y P ≤ n.
We take an open cover ω of M with each V ∈ ω having a diameter < ǫ. For every V ∈ ω consider the set V = P \p(q −1 (M\V )) and let , M), we first lift ϕ|Q (n) to a map ϕ 1 : Q (n) → Y P and define ψ : (Q (n) , Q 0 ) → M to be the map with ψ|Q 0 = ϕ|Q 0 and ψ|Q (n) = q • ϕ 1 . Obviously, ψ satisfies the required conditions. Now, we are in a position to prove the main theorem in this section.
Theorem 2.7. For a complete metric space (M, d) the following are equivalent: Proof. Implication (i) ⇒ (ii) follows from Corollary 2.4, and implica- We are going to prove this implication by induction. Since M is LC −1 (there is no such thing as a (−1)-sphere), we can suppose that M is LC n−1 and W LC n . We embed M as a closed subset of a complete metric ANR-space P and consider the following relation between the open subsets of M: V αU if V ⊂ U and every map from S k , k ≤ n, into V is null-homotopic in U , where U is any open extension of U in P . It follows from [9, Theorem 1] the existence a complete metric ρ on M generating its topology such that for ev- The metric ρ can be extended to a complete metric ρ on P , see [1]. We fix a sequence 2 k provides a metric on P . It easily seen that ̺ is a complete metric generating the topology of P . Moreover, ̺(x, y) = ρ(x, y) = ρ(x, y) for all x, y ∈ M.
For any x ∈ M and y ∈ P \W k we have and according to Lemma 2.6, there exists a map such that ψ : Q (n) ∪ Q 0 → M such that ψ|Q 0 = ϕ|Q 0 and ψ is (2 −kǫ )close to ϕ. Then the inequality 2 −kǫ < ǫ completes the proof of Claim 1.
To prove that M is LC n , we introduce the following notation: If U ⊂ M is open and γ > 0, then E γ (U) denotes any open extension of U in P witch is contained in the set B ̺ γ (M). Since M is W LC n , according to the choice of the function δ, any map ϕ : S n → M with diamϕ(S n ) < δ(ǫ) can be extended to a map ϕ : B n+1 → E γ (B ρ ǫ (ϕ(S n )), where γ > 0 is arbitrary. Now, we proceed as in the proof of Lemma 2.5 from [7]. Fix ǫ > 0 and a map f : S n → M with diamf (S n ) < δ(ǫ/2). We are going to show that f can be extended to a map f : B n+1 → M such that diamf (B n+1 ) ≤ 10ǫ. The map f will be obtained as limit of a sequence {f k : B n+1 → P }, and this sequence will be constructed by induction together with a sequence of triangulations {τ k } of B n+1 such that f k (τ (n) k ) ⊂ M for all k. To start the induction, we choose τ 1 to be B n+1 , considered as one (n+1)-dimensional simplex, and let f 1 : Denote by τ k+1 any subdivision of τ k such that diamf k (σ) < δ(2 −k ǫ)/3 for any simplex σ ∈ τ k+1 . According to Claim 1, there exists a map g k : τ Therefore, diamg k (∂σ) < δ(2 −k ǫ) for any σ ∈ τ k+1 . Hence, g k |∂σ can be extended to a map g σ k : A sequence of open covers U = (U k ) k∈N of a metric space (M, d) is called a zero-sequence if lim k →∞ meshU k = 0. For any such a sequence we define Tel(U) = k∈N N(U k ∪ U k+1 ). Here N(U k ∪ U k+1 ) is the nerve of U k ∪ U k+1 with U k and U k+1 considered as disjoint sets. For any σ ∈ Tel(U) let s(σ) = max{s : σ ∈ N(U s ∪ U s+1 )}.
We complete this section by a characterization of metrizable LC nspaces similar to the characterization of metrizable ANR-spaces provided in [12] (see also [17, Theorem 6.8.1]). Proof. Suppose M is LC n and embed (M, d) isometrically in a metric ANR-space (P, ρ) as a closed subset. According to the proof of Theorem 2.1, there are metrizable space Y P and two maps p : Y P → P and q : Y p → M such that dim Y P ≤ n + 1, p is (n + 1)-invertible and q extends the map p|p −1 (M). Using the proof of Nhu's theorem [12, Theorem 1.1] for ANR's, we can find a zero-sequence V = (V k ) k∈N of P such that any map h 0 : , where K is a subcomplex of Tel(U), extends to a map h : |K| → P such that lim k →∞ diam(h(σ k )) = 0 for any sequence {σ k } of simplex of K with s(σ k ) → ∞. Let us show that the sequence and consider the open families W k = {W (U) : U ∈ U k }, k ∈ N. We may assume that each U is a proper subset of M, so W k = ∅ for all k.
Note that W k may not cover P , but any W k covers M. Moreover, meshW k ≤ meshV k , so lim k →∞ meshW k = 0. If K is a subcomplex of Tel(U), take any map f 0 : Hence, f 0 extends to a map g : |K| → P such that lim k →∞ diam(g(σ k )) = 0 for any sequence {σ k } of simplexes of K with s(σ k ) → ∞. Finally, let f : |K (n+1) | → M be the map q• g, where g : |K (n+1) | → Y P is a lifting of g.
To prove the other implication, embed M as a closed subset of a metrizable space Z with dim Z\M ≤ n + 1 and follow the proof of implication (iii) ⇒ (i) from [12, Theorem 1.1] to obtain that M is a retract of a neighborhood W 1 of M in Z (the only difference is that in Fact 1.2 from [12] we take the cover V of W 1 \M to be of order ≤ n + 1, so the nerve N(V) is a complex of dimension ≤ n + 1). Then by [13, chapter V, Theorem 3.1], M is LC n .

UV n -maps and ALC n -spaces
In this section we provide spectral characterizations of non-metrizable ALC n -compacta and cell-like compacta. Recall that a map f : X → Y between compact spaces is said to be soft [18] if for every compactum Z, its closed subset A ⊂ Z and maps h : A → X and g : Z → Y with f • h = g|A there exists a lifting g : Z → X of g extending h.
Theorem 3.1. A compactum X is ALC n if and only if X is the limit space of a σ-complete inverse system S = {X α , p β α , α < β < τ } consisting of compact metrizable LC n -spaces X α such that all bonding projections p β α , as a well all limit projections p α , are UV n -maps.
Proof. Suppose that X is the limit space of an inverse system S = {X α , p β α , α < β < τ } such that each X α is a metric LC n -compactum and all p β α are UV n -maps. We embed X in a Tychonoff cube I B , where I = [0, 1] and the cardinality of B is equal to τ . According to Shchepin's spectral theorem [18], we can assume that B is the union of countable sets B α , α ∈ A, such that B α ⊂ B β for α < β, B γ = {B γ(k) : k = 1, 2, ..} for any chain γ(1) < γ(2) < .. with γ = sup{γ(k) : k ≥ 1}, and each p β α : X β → X α is the restriction of the projection q β α : I B β → I Bα . So, each X α = q α (X) is a subset of I Bα , where q α denotes the projection q α : I B → I Bα . We also denote q α |X by p α . Choose x 0 ∈ X and its neighborhood U ⊂ X. There exists α 0 < τ and an open set Since X α 0 is LC n , there exists a neighborhood V 0 of x α 0 = p α 0 (x 0 ) such that for all k ≤ n the inclusion j 0 : V 0 ֒→ U 0 generates trivial homomorphisms j * 0 : π k (V 0 ) → π k (U 0 ) between the homotopy groups. Let V = p −1 α 0 (V 0 ) and U be an open set in I B extending U. Choose a finite family ω = {W 1 , .., W m } of open sets from the ordinary base of I B such that W = i=m i=1 W i covers p −1 α 0 (U 0 ) and W ⊂ U. Then we can find α 1 > α 0 with q −1 Since V 1 is an ANR and V 1 is LC n (as an open subset of X α 1 ), by Proposition 2.3, there exists an open in I Bα 1 extension G of V 1 which is contained in V 1 with the following property: for every map h : Z → G, where Z is at most n-dimensional metric space, there exists a map h 1 : Z → V 1 such that h and h 1 are homotopic in V 1 . Finally, let V = q −1 α 1 (G). It is easily seen that V is an open extension of V and V ⊂ W ⊂ U . Consider a map f : S k → V , where k ≤ n. Then there exists a map g : S k → V 1 such that q α 1 • f and g are homotopic in V 1 . We are going to show that g is homotopic to a constant map in the set U 1 . This will be done if the inclusion j 1 : V 1 ֒→ U 1 generates a trivial homomorphism j * 1 : π k (V 1 ) → π k (U 1 ).
To this end, we consider the following commutative diagram: Since X α 1 is LC n and the map p α 1 α 0 is UV n , each fiber of p α 1 α 0 is an UV n -set in X α 1 , see Proposition 2.5. Then, according to [8,Theorem 5.3], both vertical homomorphisms from the above diagram are isomorphisms. This implies that j * 1 is trivial because so is j * 0 . Hence, q α 1 • f is homotopic to a constant map in the set V 1 ∪ U 1 ⊂ q α 1 (W ) (recall that q α 1 • f is homotopic to g in V 1 and g is homotopic to a constant map in U 1 ). Therefore, there is a map f 1 : B n+1 → q α 1 (W ) extending q α 1 • f . Since q α 1 is a soft map, and W = q −1 α 1 (q α 1 (W )), f can be extended to a mapf : B n+1 → W . Thus, the inclusion V ֒→ U generates a trivial homomorphism between π k ( V ) and π k ( U ) for any k ≤ n. So, X is ALC n . Now, suppose X is ALC n , and consider X as a subset of some I B . Since the sets V and V in the ALC n definition depend on the point x, the set U and its open extension U, respectively, we use the notations λ(x, U) = V and λ(x, U, U) = V . First, we show that the sets V and V can be chosen to be functionally open in X and in I B , respectively. Indeed, if U ⊂ X is a neighborhood of x ∈ X, we take a functionally open in X neighborhood V * of x with V * ⊂ λ(x, U). Then for a given open in I B extension U of U and every y ∈ V * choose a functionally open in I B neighborhood G(y) of y with G(y) ⊂ λ(x, U, U) ∩ G, where G is an open in I B extension of V * . Since V * , as a functionally open subset of X, is Lindelöf, there exist countably many sets G(y i ) whose union covers V * . Obviously the set G = ∞ i=1 W (y i ) is a functionally open in I B extension of V * which is contained in λ(x, U, U). So, every map from S k to G, k ≤ n, is homotopic in U to a constant map.
Therefore, for every open set U ⊂ X and every x ∈ U there exists a functionally open in X set V = λ(x, U) such that for any open in I B extension U of U we can find a functionally open in I B extension λ(x, U, U) of V contained in U with all homomorphisms π k (λ(x, U, U) → π k ( U ), k ≤ n, being trivial. If U is functionally open in X, then it is Lindelöf and there are countably many x i ∈ U such that {λ(x i , U) : i = 1, 2, ..} is a cover of U. We fix such a countable cover γ(U) for any functionally open set U ⊂ X.
Let A ⊂ B and W 0 , W 1 , .., W k be elements of the standard open base . We say that a set A ⊂ B is admissible if the following holds: ., W k ] A ) and all finitely many elements W 0 , W 1 , .., W k of B A satisfying condition (w); Recall that for any functionally open set U in I B (resp., in X) there is a countable set s(U) ⊂ B such that q −1 s(U ) (q s(U ) (U)) = U (resp., p −1 s(U ) (p s(U ) (U)) = U). Claim 2. For any set A ⊂ B there exists an admissible set C ⊂ B of cardinality |A|.ℵ 0 containing A.
We construct by induction sets and all finitely many W 0 , W 1 , .., W m ∈ B A k satisfying condition (w). The construction follows from the fact that the cardinality of each base B A k is |A|.ℵ 0 and the families γ(q −1 A k (W ) ∩ X) and γ([W 0 , W 1 , .., W m ] A k ) are countable provided W, W 0 , .., W k ∈ B A k . It is easily seen that the set C = ∞ k=1 A k is as required. Claim 3. X A is an ALC n -space for every admissible set A ⊂ B. Let y ∈ U, where U is open in X A . Take x ∈ X and W 0 ∈ B A containing y such that W 0 ∩ X A ⊂ U and q A (x) = y. Then x belongs to some V x ∈ γ(q −1 is a functionally open in X A neighborhood of y, which is contained in U. Take any open extension U ⊂ I A of U, and finitely many W 1 , .., is a functionally open in I A extension of V . We are going to show that all homomorphisms π k ( V ) → π k ( U), k ≤ n, are trivial. Indeed, every map f : S k → V can be lifted to a map g : Then y belongs to some V y ∈ γ(q −1 Then, as in the proof of Claim 3, we can show that the inclusion V ֒→ U generates trivial homomorphisms π k (V ) → π k (U). Hence, F is UV n .
Claim 5. The union of any increasing sequence of admissible subsets of B is also admissible.
This claim follows directly from the definition of admissible sets. Now we can complete the proof of Theorem 3.1. According to Claim 2 and Claim 5, the set B is covered by a family S of countable sets such that S is stable with respect to countable unions. Then, by Claim 3, each X A , A ∈ S, is a metric ALC n -compactum . Hence, Proposition 2.7 yields that all spaces X A , A ∈ S, are LC n . Moreover, the projections p A 1 A 2 are UV n -maps for any A 1 , A 2 ∈ S with A 2 ⊂ A 1 . Because the set B is admissible, it follows from Claim 4 that the limit projections p A : X → X A , A ∈ S, are also UV n -maps. Therefore, X is limit space of the σ-complete inverse system {X A , p A 1 A 2 , A, A 1 , A 2 ∈ S}. Corollary 3.2. Any LC n -compactum X is an ALC n -space.
Proof. We embed X in some I B and let A 0 ⊂ B be a countable set. According to the factorization theorem of Bogatyi-Smirnov [2, Theorem 3], there is a metric compactum Y 1 and maps g 1 : X → Y 1 , h 1 : Y 1 → X A 0 such that p A 0 = h 1 • g 1 and all fibers of g 1 are UV n -sets in X. Then, by [8,Theorem 5.4], Y 1 is LC n . Since g 1 depends on countably many coordinates, there is a countable set A 1 ⊂ B containing A 0 and a map f 1 : In this way we construct countable sets A k ⊂ A k+1 ⊂ B and LC n metric compacta Y k together with maps g k : X → Y k , f k : : and the fibers of each g k are UV nsets in X. Let A be the union of all A k . Then X A is the limit space of the inverse sequence S = {Y k , s k+1 k = f k • h k+1 }. According to [8,Theorem 5.3], for all open sets U ⊂ Y k the group π m (U) is isomorphic to π m (g −1 k (U)), m = 0, 1, .., n. This property of the maps g k implies that any s k+1 k : Y k+1 → Y k is an UV n -map. Hence, by Theorem 3.1, X A is an ALC n -compactum (as the limit of an inverse sequence of metric LC n -compacta and bounding UV n -maps). Finally, by Theorem 2.7, X A is also an LC n -space. Moreover, for any y ∈ X A we have p −1 A (y) = k≥1 g −1 k (y k ), where y k = s k (y) with s k : X A → Y k being the projections of S. Because all g −1 k (y k ) are UV n -sets in X, so is the set p −1 A (y). Therefore, every countable subset A 0 of B is contained in an element of the family A consisting of all countable sets A ⊂ B such that X A is LC n and the fibers of the map p A are UV n -sets in X. It is easily seen that the union of an increasing sequence of elements of A is again from A, and that p C A : X C → X A is an UV n -map for all A, C ∈ A with A ⊂ C. So, the inverse system {X A , p C A , A, C ∈ A} is σ-complete and consists of metric LC n -compacta and UV n -bounding maps. Then by Theorem 3.1, X is ALC n (observe that the proof of the "if" part of Theorem 3.1 does not need the assumption that all limit projections are UV n -maps). Theorem 3.1 shows that the class of ALC n -compacta is adequate to the class of UV n -maps. Next theorem provides another classes of compacta adequate to all continuous maps. Theorem 3.3. A compactum X is a cell-like (resp., UV n ) space if and only if X is the limit space of a σ-complete inverse system consisting of cell-like (resp., UV n ) metric compacta.
Proof. Suppose X is a cell-like compactum. Because this means that X has a shape of a point, we can apply Corollary 8.4.8 from [3] stating that if ϕ is a shape isomorphism between the limit spaces of two σcomplete inverse systems {X α , p β α , α ∈ A} and {Y α , q β α , α ∈ A} of metric compacta, then the set of those α ∈ A for which there exist shape isomorphisms ϕ α : X α → Y α satisfying Sh(q α ) • ϕ = ϕ α • Sh(p α ) is cofinal and closed in A. So, according to this corollary, X is the limit space of a σ-complete inverse system consisting of metric cell-like compacta. In case X is an UV n -compactum, it has an n-shape of a point (this notion was introduced by Chigogidze in [4]), and the above arguments apply.
Suppose now that X is the limit space of a σ-complete inverse system {X α , p β α , α ∈ A} such that all X α are metric cell-like compacta. As in the proof of Theorem 3.1, we can embed X in a Tychonoff cube I B , where B is the union of countable sets B α , α ∈ A, such that B α ⊂ B β for α < β, B γ = {B γ(k) : k = 1, 2, ..} for any chain γ(1) < γ(2) < .. with γ = sup{γ(k) : k ≥ 1}, and each p β α : X β → X α is the restriction of the projection q β α : I B β → I Bα . Then X α = q α (X) ⊂ I Bα with q α being the projection from I B onto I Bα . If U is a neighborhood of X in I B , there is α and an open set U α in I Bα such that q −1 α (U α ) ⊂ U. Since X α is a cell-like space, there exists a closed neighborhood V α ⊂ I Bα of X α contractible in U α . Using that q α is a soft map, we conclude that q −1 α (V α ) is contractible in q −1 α (U α ). Similarly, we can show that any limit space of a σ-complete inverse system of metric UV n -compacta is also an UV n -compactum.