����������� ������ 52 (2000), 83–97

UDK 515.122.2

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RELATIONSHIPS BETWEEN USUAL AND APPROXIMATE INVERSE SYSTEMS

Ivan Lončar

Abstract. We shall prove that if X = {X_{a}, p_{ab}, A} is an approximate inverse system of compact non-metric spaces with surjective bonding mappings p_{ab} such that each X_{a} is a limit of a usual τ-directed inverse system X(a) = {X_{(a,γ)}, f_{(a,γ)(a,δ)}, Γ_{a}} of metric compact spaces, then there exist: 1) a usual τ-directed inverse system X_{D} = {X_{d}, F_{de}, D} whose inverse limit X_{D} is homeomorphic to X = lim X, 2) every X_{d} is a limit of an approximate inverse system {X_{(a,γa)}, g_{(a,γa)(b,γb)}, A} of compact metric spaces X_{(a,γa)}, 3) if the mappings p_{ab} and f_{(a,γ)(a,δ)} are monotone, then g_{(a,γa)(b,γb)} and F_{de} are monotone.

1. Introduction

In this paper we shall use the notion of inverse systems X = {X_{a}, p_{ab}, A} and their limits in the usual sense [1, p. 135].

The cardinality of a set X will be denoted by card(X). The cofinality of a cardinal number m will be denoted by cf(m). Cov(X) is the set of all normal coverings of a topological space X. If U, V ∈ Cov(X) and V refines U, we write V ≤ U. For two mappings f, g: Y → X which are U-near (for every y ∈ Y there exists a U ∈ U with f(y), g(y) ∈ U), we write (f, g) ≤ U. A basis of (open) normal coverings of a space X is a collection C of normal coverings such that every normal covering U ∈ Cov(X) admits a refinement V ∈ C. We denote by cw(X) (covering weight) the minimal cardinal of a basis of normal coverings of X [9, p. 181].

Lemma 1. [9, Example 2.2] If X is a compact Hausdorff space, then cw(X) = w(X).

The notion of approximate inverse system X = {X_{a}, p_{ab}, A} will be used in the sense of S. Mardešić [11].

Definition 1. An approximate inverse system is a collection X = {X_{a}, p_{ab}, A}, where (A, ≤) is a directed preordered set, X_{a}, a ∈ A, is a topological space and p_{ab}: X_{b} → X_{a}, a ≤ b, are mappings such that p_{aa} = id and the following condition (A2) is satisfied:

AMS Subject Classification: 54 C 10, 54 F 15, 54 B 35

Keywords and phrases: Approximate inverse system, usual inverse system.

(A2) For each a ∈ A and each normal cover U ∈ Cov(X_{a}) there is an index b ≥ a

such that (p_{ac}p_{cd}, p_{ad}) ≤ U, whenever a ≤ b ≤ c ≤ d. An approximate map p = {p_{a} : a ∈ A}: X → X into an approximate system

X = {X_{a}, p_{ab}, A} is a collection of maps p_{a}: X → X_{a}, a ∈ A, such that the following condition holds (AS) For any a ∈ A and any U ∈ Cov(X_{a}) there is b ≥ a such that (p_{ac}p_{c}, p_{a}) ≤ U

for each c ≥ b. (See [10]). Let X = {X_{a}, p_{ab}, A} be an approximate system and let p = {p_{a} : a ∈

A}: X → X be an approximate map. We say that p is a limit of X provided it has the following universal property: (UL) For any approximate map q = {q_{a} : a ∈ A}: Y → X of a space Y there

exists a unique map g: Y → X such that p_{a}g = q_{a}. Let X = {X_{a}, p_{ab}, A} be an approximate system. A point x = (x_{a}) ∈ ∏{X_{a} :

a ∈ A} is called a thread of X provided it satisfies the following condition:

(L) (∀a ∈ A)(∀U ∈ Cov(X_{a}))(∃b ≥ a)(∀c ≥ b) p_{ac}(x_{c}) ∈ st(x_{a}, U).

If X_{a} is a T_{3.5} space, then the sets st(x_{a}, U), U ∈ Cov(X_{a}), form a basis of the topology at the point x_{a}. Therefore, for an approximate system of Tychonoff spaces condition (L) is equivalent to the following condition: (L)^{∗} (∀a ∈ A) lim{p_{ac}(x_{c}) : c ≥ a} = x_{a}.

Let τ be an infinite cardinal. We say that a partially ordered set A is τ-directed if for each B ⊆ A with card(B) ≤ τ there is an a ∈ A such that a ≥ b for each b ∈ B. If A is ℵ_{0}-directed, then we will say that A is σ-directed. An inverse system X = {X_{a}, p_{ab}, A} is said to be τ-directed if A is τ-directed. An inverse system X = {X_{a}, p_{ab}, A} is said to be σ-directed if A is σ-directed.

The proof of the following theorem is similar to the proof of Theorem 1.1 of [4].

Theorem 1. Let X = {X_{a}, p_{ab}, A} be a σ-directed approximate inverse system of compact spaces with surjective bonding mappings and limit X. Let Y be a metric compact space. For each surjective mapping f: X → Y there exists an a ∈ A such that for each b ≥ a there exists a mapping g_{b}: X_{b} → Y such that f = g_{b}p_{b}.

Theorem 2. Let X be a compact space. There exists a σ-directed inverse system X = {X_{a}, p_{ab}, A} of compact metric spaces X_{a} and surjective bonding mappings p_{ab} such that X is homeomorphic to lim X.

Proof. It is well-known that there exists a usual inverse system Y = {Y_{α}, q_{αβ}, Σ} of metric spaces Y_{α} and surjective bonding mappings such that X is homeomorphic to lim Y. By Theorem 9.5 of [12] there exists a σ-directed inverse system X = {X_{a}, p_{ab}, A} such that lim X is homeomorphic to lim Y and each X_{a} is the limit of a countable inverse subsystem of Y. This means that each X_{a} is a metric compact space.

Theorem 3. [8, p. 163, Theorem 2.] If X is a locally connected compact space, then there exists an inverse system X = {X_{a}, p_{ab}, A} such that each X_{a} is a metric locally connected compact space, each p_{ab} is a monotone surjection and X is homeomorphic to lim X. Conversely, the inverse limit of such system is always a locally connected compact space.

Remark 1. We may assume that X = {X_{a}, p_{ab}, A} in Theorem 3 is σ-directed [12, Theorem 9.5].

Theorem 4. [13, Corollary 2.9] If X is a hereditarily locally connected continuum, then there exists a σ-directed inverse system X = {X_{a}, p_{ab}, A} such that each X_{a} is a metrizable hereditarily locally connected continuum, each p_{ab} is a monotone surjection and X is homeomorphic to lim X.

Theorem 5. [3, Corollary 3] Let X = {X_{a}, p_{ab}, A} be a σ-directed inverse system of hereditarily locally connected continua X_{a}. Then X = lim X is hereditarily locally connected.

The following theorem is Theorem 1.7 from [5].

Theorem 6. Let X = {X_{a}, p_{ab}, A} be a σ-directed inverse system of compact metrizable spaces and surjective bonding mappings. Then X = lim X is metrizable if and only if there exists an a ∈ A such that p_{b}: X → X_{b} is a homeomorphism for each b ≥ a.

2. Approximate subsystems

In this Section we investigate the approximate subsystem of an approximate system X = {X_{a}, p_{ab},A}. We start with the following definition.

Definition 2. Let X = {X_{a}, p_{ab}, A} be an approximate inverse system and let B be a directed subset of A such that {X_{b}, p_{bc}, B} is an approximate inverse system. We say that {X_{b}, p_{bc}, B} is an approximate subsystem of X = {X_{a}, p_{ab}, A} if there exists a mapping q: lim X → lim{X_{b}, p_{bc}, B} such that

p_{b}q = P_{b}, b ∈ B,

where p_{b}: lim{X_{b}, p_{bc}, B} → X_{b} and P_{b}: lim X → X_{b}, b ∈ B, are natural projections.

We say that an approximate system X = {X_{a}, p_{ab}, A} is irreducible if for each B ⊂ A with card(B) < card(A) it follows that B is not cofinal in A.

Lemma 2. Let X = {X_{a}, p_{ab}, A} be an approximate inverse system. There exists a cofinal subset B of A such X = {X_{a}, p_{ab}, B} is irreducible.

Proof. Consider the family B of all cofinal subsets of B of A. The set {card(B) : B ∈ B} has the minimal element b since each card(B) is some initial ordinal number. Let B ∈ B be such that card(B) = b. It is clear that {X_{a}, p_{ab}, B} is irreducible.

In the sequel we will assume that X = {X_{a}, p_{ab}, A} is irreducible.

Lemma 3. Let X = {X_{a}, p_{ab}, A} be an approximate inverse system of compact spaces such that card(A) = ℵ_{0}. Then there exists a countable well-ordered subset B of A such that the collection {X_{b}, p_{bc}, B} is an approximate inverse sequence and lim X is homeomorphic to lim{X_{b}, p_{bc}, B}.

Proof. Let ν be any finite subset of A. There exists a δ(ν) ∈ A such that δ ≤ δ(ν) for each δ ∈ ν. Since A is infinite, there exists a sequence {ν_{n} : n ∈ ℕ} such that ν_{1} ⊆ … ⊆ ν_{n} ⊆ ⋯ and A = ⋃{ν_{n} : n ∈ ℕ}. Recursively, we define the sets A_{1},…, A_{n},… by

A_{1} = ν_{1} ∪ {δ(ν_{1})},

and

A_{n+1} = A_{n} ∪ ν_{n+1} ∪ {δ(A_{n} ∪ ν_{n+1})}.

It follows that there exists a sequence

A_{1} ⊆ A_{2} ⊆ … ⊆ A_{n} ⊆ ⋯

of finite sets A_{n} such that A = ⋃{A_{n} : n ∈ ℕ}. Let b_{1} = δ(A_{1}) and b_{n} ≥ δ(A_{n}), b_{n−1} if n ≥ 2. We obtain a sequence B = {b_{n} : n ∈ ℕ} such that B is cofinal in A. By virtue of [10, Theorem (1.19)] it follows that lim X is homeomorphic to lim{X_{b}, p_{bc}, B}.

Now we consider irreducible approximate inverse systems X = {X_{a}, p_{ab}, A} with card(A) ≥ ℵ_{1}.

Lemma 4. Let A be a directed set. For each subset B of A there exists a directed set F_{∞}(B) such that card(F_{∞}(B)) = card(B).

Proof. For each B ⊆ A there exists a set F_{1}(B) = B ⋃{δ(ν) : ν ∈ B}, where ν is a finite subset of B and δ(ν) is defined as in the proof of Lemma 3. Put

F_{n+1} = F_{1}(F_{n}(B),

and

F_{∞}(B) = ⋃{F_{n}(B) : n ∈ ℕ}.

It is clear that

F_{1}(B) ⊆ F_{2}(B) ⊆ … ⊆ F_{n}(B) ⊆ ⋯

The set F_{∞}(B) is directed since each finite subset ν of F_{∞}(B) is contained in some F_{n}(B) and, consequently, δ(ν) is contained in F_{∞}(B).

If B is finite, then card(F_{∞}(B)) = ℵ_{0}. If card(B) ≥ ℵ_{0}, then we have card({δ(ν) : ν ∈ B}) ≤ card(B)ℵ_{0}. We infer that card(F_{1}(B)) ≤ card(B)ℵ_{0}. Similarly, card(F_{n}(B)) ≤ card(B)ℵ_{0}. This means that card(F_{∞}(B)) ≤ card(B)ℵ_{0}. Thus

card(F_{∞}(B)) ≤ card(B)ℵ_{0},  if  card(B) < card(A).

The proof is completed.

Lemma 5. Let {X_{a}, p_{ab}, A} be an approximate inverse system such that cw(X_{a}) < card(A), a ∈ A. For each subset B of A with card(B) < card(A), there exists a directed set G_{∞}(B) ⊇ B such that the collection {X_{a}, p_{ab}, G_{∞}(B)} is an approximate system and card(G_{∞}(B)) = card(B).

Proof. Let B_{a} be a base of normal coverings of X_{a}. Let U_{a} ∈ B_{a}. By virtue of (A2) there exists an a(U_{a}) ∈A such that (p_{ad}, p_{ac}p_{cd}) ≤ U_{a}, a ≤ a(U_{a}) ≤ c ≤ d. For each subset B of A we define G_{∞}(B) by induction as follows:

a) Let G_{1}(B) = F_{∞}(B). From Lemma 4 it follows that card(G_{1}(B)) = card(F_{∞}(B)) = card(B).

b) For each n > 1 we define G_{n}(B) as follows:

1) If n is odd then G_{n}(B) = F_{∞}(G_{n−1}(B)),

2) If n is even, then G_{n}(B) = G_{n−1}(B) ∪ {a(U_{a}) : U_{a} ∈ B_{a}, a ∈ G_{n−1}(B)}. Since card(B_{a}) < card(A) the set G_{n}(B) has the cardinality < card(A). Now we define G_{∞}(B) = ⋃{G_{n}(B) : n ∈ ℕ}. It is obvious that card(G_{∞}(B)) < card(A).

The set G_{∞}(B) is directed. Let a, b be a pair of elements of G_{∞}(B). There exists an n ∈ ℕ such that a, b ∈ G_{n}(B). We may assume that n is odd. Then a, b ∈ F_{∞}(G_{n−1}(B)). Thus there exists a c ∈ F_{∞}(G_{n−1}(B)) such that c ≥ a, b. It is clear that c ∈ G_{∞}(B). The proof of directedness of G_{∞}(B) is completed.

The collection {X_{a}, p_{ab}, G_{∞}(B)} is an approximate system. It suffices to prove that the condition (A2) is satisfied. Let a be any member of G_{∞}(B). There exists an n ∈ ℕ such that a ∈ G_{n}(B). We have two cases.

1) If n is odd then G_{n}(B) = F_{∞}(G_{n−1}(B)). This means that a ∈ F_{∞}(G_{n−1}(B)). By definition of F_{∞}(G_{n−1}(B)) we infer that a(U_{a}) ∈ F_{∞}(G_{n−1}(B)). Thus (A2) is satisfied.

2) If n is even, then G_{n}(B) = G_{n−1}(B) ∪ {a(U_{a}) : U_{a} ∈ Cov(X_{a}), a ∈ G_{n−1}(B)}. In this case a ∈ G_{n+1}(B) ⊆ G_{∞}(B). Arguing as in the case 1, we infer that (A2) is satisfied.

Theorem 7. Let X = {X_{a}, p_{ab}, A} be an approximate inverse system of compact spaces. If λ ≤ w(X_{a}) ≤ τ < card(A) for each a ∈ A, then lim X is homeomorphic to a limit of a λ-directed usual inverse system {X_{α}, q_{αβ}, T }, where each X_{α} is a limit of an approximate inverse subsystem {X_{γ}, p_{αβ}, Φ}, card(Φ) = λ.

Proof. The proof consists of several steps.

Step 1. Let B = {B_{μ} : μ ∈ M} be a family of all subsets of A with card(B_{α}) = λ. Put A_{μ} = G_{∞}(B_{μ}) (Lemma 5) and let Δ = {A_{μ} : μ ∈ M} be ordered by inclusion ⊆.

Step 2. If Φ and Ψ are in Δ such that Φ ⊂ Ψ, then there exists a mapping q_{ΦΨ} : lim{X_{α}, p_{αβ}, Ψ} → lim{X_{γ}, p_{αβ}, Φ}.

Namely, if x = (x_{α}, α ∈ Ψ) ∈ lim{X_{α}, p_{αβ}, Ψ}, then by definition of the threads of {X_{α}, p_{αβ}, Ψ} the condition (L) is satisfied. If (L) is satisfied for x = (x_{α}, α ∈ Ψ) ∈ lim{X_{α}, p_{αβ}, Ψ}, then it is satisfied for (x_{γ}, γ ∈ Φ) since the required a′ in (L) lies—by definition of the set Φ—in the set Φ. This means that (x_{γ}, γ ∈ Φ) ∈ lim{X_{γ}, p_{αβ}, Φ}. Now we define q_{ΦΨ}(x) = (x_{γ}, γ ∈ Φ).

Step 3. The collection {X_{Φ}, q_{ΦΨ}, Δ} is a usual inverse system. It suffices to prove transitivity, i.e., if Φ ⊆ Ψ ⊆ Ω, then q_{ΦΨ}q_{ΨΩ} = q_{ΦΩ}. This easily follows from the definition of q_{ΦΨ}.

Step 4. The space lim X is homeomorphic to lim{X_{Ψ}, q_{ΦΨ}, Δ}, where X_{Φ} = lim{X_{γ}, p_{αβ}, Φ}. We shall define a homeomorphism H: lim X → lim{X_{Ψ}, q_{ΦΨ}, Δ}. Let x = (x_{a} : a ∈ A) be any point of lim X. Each collection {x_{a} : a ∈ Φ ∈ Δ} is a point x_{Φ} of X_{Φ} since X_{Φ} = lim{X_{a}, p_{ab}, Φ}. Moreover, from the definition of q_{ΦΨ} (Step 2) it follows that q_{ΦΨ}(x_{Ψ}) = x_{Φ}, Ψ ⊇ Φ. Thus, the collection {x_{Φ} : Φ ∈ Δ} is a point of lim{X_{Φ}, q_{ΦΨ}, Δ}. Let H(x) = {x_{Φ}, Φ ∈ Δ}. Thus, H is a continuous mapping of lim X to lim{X_{Ψ}, q_{ΦΨ}, Δ}. In order to complete the proof it suffices to prove that H is 1–1 and onto. Let us prove that H is 1–1. Let x = (x_{a} : a ∈ A) and y = (y_{a} : a ∈ A) be a pair of points of lim X. This means that there exists an a ∈A such that y_{a} ≠ x_{a}. There exists a Φ ∈ Δ such that a ∈ Φ. Thus, the collections {x_{a} : a ∈ Φ} and {y_{a} : a ∈ Φ} are different. From this we conclude that x_{Φ} ≠ y_{Φ}, x_{Φ}, y_{Φ} ∈ X_{Φ} = lim{X_{a}, p_{ab}, Φ}. Hence H is 1–1. Let us prove that H is onto. Let y = (y_{Φ} : Φ ∈ Δ) be any point of lim{X_{Ψ}, q_{ΦΨ}, Δ}. Each y_{Φ} is a collection {x_{a} : a ∈ Φ} and if Ψ ⊇ Φ, then the collection {x_{a} : a ∈ Φ} is the restriction of the collection {x_{a} : a ∈ Ψ} on Φ. Let x be the collection which is the union of all collections {x_{a} : a ∈ Φ}, Φ ∈ Δ. Hence x is a collection (x_{a} : a ∈ A) which is a point of lim X and H(x) = y.

Step 5. Inverse system {X_{Φ}, q_{ΦΨ}, Δ} is a λ-directed inverse system. Let {{X_{γ}, p_{αβ}, Φ_{κ}} : κ ≤ λ} be a collection of approximate subsystems {X_{γ}, p_{αβ}, Φ_{κ}}. The set Φ = ⋃{Φ_{κ} : κ ≤ λ} has the cardinality ≤ λ since card(Φ_{κ}) ≤ λ. By virtue of Steps 1–4 there exists an approximate subsystem {X_{γ}, p_{αβ}, Φ}, card(Φ) = λ. This means that {X_{Φ}, q_{ΦΨ}, Δ} is a λ-directed inverse system.

Corollary 1. Let X = {X_{a}, p_{ab}, A} be an approximate inverse system of compact metric spaces. Then lim X is homeomorphic to the limit of a σ-directed usual inverse system {X_{α}, q_{αβ}, Δ}, where each X_{α} is a limit of an approximate inverse subsystem {X_{γ}, p_{αβ}, Φ}, card(Φ) = ℵ_{0}.

Lemma 6. Let X = {X_{a}, p_{ab}, A} be an approximate system such that X_{a}, a ∈ A, are compact locally connected spaces and p_{ab} are monotone surjections. If Y = {X_{b}, p_{cd}, B} is an approximate subsystem of X, then the mapping q_{AB}: lim X → lim Y (defined in Step 2 of the proof of Theorem 7) is a monotone surjection.

Proof. Let P_{a}: lim X → X_{a}, a ∈ A, be the natural projection. Similarly, let p_{a}: lim Y → X_{a}, a ∈ B, be the natural projection. From the definition of q_{AB} (Step 2 of the proof of Theorem 7) it follows that p_{a}q_{AB} = P_{a} for each a ∈ B. By virtue of [10, Corollary 4.5] and [7, Corollary 5.6] it follows that P_{a} and p_{a} are monotone surjections. Let us prove that q_{AB} is a surjection. Let y = (y_{a} : a ∈ B) ∈ lim Y. The sets P_{a}^{−1}(y_{a}), a ∈ B, are non-empty since P_{a} is surjective for each a ∈ A. From the compactness of lim X it follows that a limit superior Z = Ls{P_{a}^{−1}(y_{a}), a ∈ B} is a non-empty subset of lim X. We shall prove that for each z = (z_{a} : a ∈ A) ∈ Z, P_{a}(z) = y_{a}. Suppose that P_{a}(z) ≠ y_{a}. There exists a pair U, V of open disjoint subsets of X_{a} such that y_{a} ∈ U and P_{a}(z) ∈ V. For sufficiently large b ∈ B, P_{a}(P_{b}^{−1}(_{b})) is in U because of (AS). This means that P_{a}^{−1}(V) ∩ P_{b}^{−1}(y_{b}) = ∅ for sufficiently large b ∈ B. This contradicts the assumption z ∈ Ls{P_{a}^{−1}(y_{a}), a ∈ B}. Hence q_{AB} is a surjection. In order to complete the proof it suffices to prove that q_{AB} is monotone. Take a point y ∈ lim Y and suppose that q_{AB}^{−1}(y) is disconnected. There exists a pair U, V of disjoint open sets in lim X such that q_{AB}^{−1}(y) ⊆ U ∪ V. From the compactness of lim X it follows that q_{AB} is closed. This means that there exists an open neighborhood W of y such that q_{AB}^{−1}(y) ⊆ q_{AB}^{−1}(W) ⊆ U ∪ V. From the definition of the basis in lim Y it follows that there exists an open set W_{a} in some X_{a}, a ∈B such that y ∈ p_{a}^{−1}(W_{a}) ⊆ W. Moreover, we may assume that W_{a} is connected since X_{a} is locally connected. Then P_{a}^{−1}(W_{a}) is connected since P_{a} is monotone [7, Corollary 5.6]. Moreover, q_{AB}^{−1}(y) ⊆ P_{a}^{−1}(W_{a}) and P_{a}^{−1}(W_{a}) ⊆ U ∪ V since P_{a} = p_{a}q_{AB}. This is impossible since U and V are disjoint open sets and P_{a}^{−1}(W_{a}) is connected.

Theorem 8. Let X = {X_{a}, p_{ab}, A} be an approximate inverse system of compact spaces. If λ ≤ w(X_{a}) < card(A) for each a ∈ A and cf(card(A)) ≠ λ, then X = lim X is homeomorphic to a limit of a λ-directed usual inverse system {X_{α}, q_{αβ}, T }, where each X_{α} is a limit of an approximate inverse subsystem {X_{γ}, p_{αβ}, Φ}, card(Φ) = λ. Moreover, if card(A) is a regular cardinal, then X = lim X is homeomorphic to a limit of a λ-directed usual inverse system {X_{α}, q_{αβ}, T }, where each X_{α} is a limit of an approximate inverse subsystem {X_{γ}, p_{αβ}, Φ}, card(Φ) = λ.

A directed preordered set (A, ≤) is said to be cofinite provided each a ∈ A has only finitely many predecessors. If a ∈ A has exactly n predecessors, we shall write p(a) = n + 1. Hence, a ∈ A is the first element of (A, ≤) if and only if p(a) = 1.

Lemma 7. If (A, ≤) is cofinite, then it satisfies the following principle of induction:

Let B ⊂ A be a set such that:

(i) B contains all the first elements of A, (ii) if B contains all the predecessors of a ∈ A, then a ∈ B.

Then B = A.

Lemma 8. [15, Lemma 1] Let q = (q_{a}): Y → Y = {Y_{b}, V_{b}, q_{ab′} , B} be an approximate map (approximate resolution) of a space Y. Then there exists an approximate map (approximate resolution) q = (q_{a}): Y → Y = {Yc^{′}, Vc^{′}, q_{cc′} , C} of the space Y and an increasing surjection t: C → B satisfying the following conditions:

(i) C is directed, unbounded, antisymmetric and cofinite set; (ii) (∀c ∈ C)(∀b ∈ B)(∃c′ > c) t(c′) > b; (iii) (∀c ∈ C) Y′_{c} = Y_{t(c)}, V′_{c} = V_{t(c)}, q′_{c} = q_{t(c)} and q′_{cc′} = q_{t(c)t(c′)}, whenever c < c′.

Corollary 2. Let X = {X_{a}, p_{ab}, A} be an approximate inverse system of compact spaces. Then there exists a cofinite approximate inverse system Y = {Y_{c}, p_{cc′}, C} such that each Y_{c} is some X_{a}, each p_{cc′} is some p_{ab} and lim X is homeomorphic to lim Y.

Proof. By virtue of Theorem (4.2) of [10] an approximate map p: X → X is an approximate resolution if and only if it is a limit of X = {X_{a}, p_{ab}, A}. Apply Lemma 8.

Theorem 9. Let X = {X_{a}, p_{ab}, A} be an approximate inverse system of compact non-metric spaces with surjective bonding mappings p_{ab}. If each X_{a} is a limit of a usual σ-directed inverse system X(a) = {X_{(a,γ)}, f_{(a,γ)(a,δ)}, Γ_{a}} of metric compact spaces, then:

1. there exists a usual σ-directed inverse system X_{D} = {X_{d}, F_{de}, D} whose inverse limit X_{D} is homeomorphic to X = lim X,

2. every X_{d} is a limit of an approximate inverse system {X_{(a,γ_{a})}, g_{(a,γ_{a})(b,γ_{b})}, A} of compact metric spaces X_{(a,γ_{a})},

3. if the mappings p_{ab} and f_{(a,γ)(a,δ)} are monotone, then g_{(a,γ_{a})(b,γ_{b})} and F_{de} are monotone. Proof. The proof consists of several steps. In the Steps 0.–11. we shall define a

usual inverse system X_{D} = {X_{d}, F_{de}, D} whose inverse limit X_{D} is homeomorphic to X = lim X.

Step 0. From Corollary 2 it follows that we may assume that A is cofinite. Step 1. For each X_{a} there exists a σ-directed inverse system

X(a) = {X_{(a,γ)}, f_{(a,γ)(a,δ)}, Γ_{a}}  (1)

such that each X_{(a,γ)} is a metric compact space, each f_{(a,γ)(a,δ)} is monotone and surjective and X_{a} is homeomorphic to lim X(a). Now we have the following diagram

\begin{matrix}X_{a} & ⟵^{p_{ab}} & X_{b} & ⟵^{p_{bc}} & X_{c} & ⟵^{p_{d}} & X \\ ↓^{f_{(a,γa)}} & ↓^{f_{(b,γ_{b})}} & ↓^{f_{(c,γc)}} \\ X_{(a,γ_{a})} & X_{(b,γ_{b})} & X_{(c,γ_{c})} \\ ↓^{f_{(a,γa)(a,δa)}} & ↓^{f_{(b,γ_{b})(b,δ_{b})}} & ↓^{f_{(c,γc)(c,δc)}} \\ X_{(a,δ_{a})} & X_{(b,δ_{b})} & X_{(c,δ_{c})} \\ ↓ & ↓ & ↓ \end{matrix}  (2)

Step 2. Put B = {(a, γ_{a}) : a ∈ A, γ_{a} ∈ Γ_{a}} and put C to be the set of all subsets c of B of the form

c = {(a, γ_{a}) : a ∈ A},  (3)

where every γ_{a} is the fixed element of Γ_{a}.

Step 3. Let D be a subset of C containing all c ∈ C for which there exist the mappings

g_{(a,γ_{a})(b,γ_{b})} : X_{(b,γ_{b})} → X_{(a,γ_{a})},  b ≥ a,  (4)

such that

{X_{(a,γ_{a})}, g_{(a,γ_{a})(b,γ_{b})}, A}  (5)

is an approximate inverse system and each diagram

\begin{matrix}X_{a} & ⟵^{p_{ab}} & X_{b} \\ ^{f_{(a,γa)}}↓ & & ↓^{f_{(b,γ_{b})}} \\ X_{(a,γ_{a})} & ⟵_{g_{(a,γa)(b,γ_{b})}} & X_{(b,g_{b})} \end{matrix}  (6)

commutes, where f_{(a,γ_{a})} : X_{a} → X_{(a,γ_{a})} is the canonical projection. It is clear that the mapping g_{(a,γ_{a})(b,γ_{b})} is unique since f_{(b,γ_{b})} is a surjection.

Step 4. The set D is non-empty. Moreover, for each subset S_{a} ⊂ Γ_{a}, a ∈ A, card(S_{a}) ≤ ℵ_{0}, there exists a d ∈ D such that d = {(a, γ_{a}) : a ∈ A}, γ_{a} ≥ γ for every γ ∈ S_{a}. Let a ∈ A be some first element of A and let γ_{a} ∈ Γ_{a} such that γ_{a} ≥ γ for every γ ∈ S_{a}. The space X_{(a,γ_{a})} is a metric compact space and there exist mappings f_{(a,γ_{a})}p_{ab}: X_{b} → X_{(a,γ_{a})}, b ≥ a. By virtue of Theorem 1 for each b ≥ a there exists a γ_{b}^{1} ∈ Γ_{b} such that for each γ_{b} ≥ γ_{b}^{1}, γ, where γ ∈ S_{b}, there exists a monotone surjective mapping g_{(a,γ_{a})(b,γ_{b})} : X_{(b,γ_{b})} → X_{(a,γ_{a})} with f_{(a,γ_{a})}p_{ab} = g_{(a,γ_{a})(b,γ_{b})}f_{(b,γ_{b})}, i.e., the diagram

\begin{matrix}X_{a} & ⟵^{p_{ab}} & X_{b} \\ ^{f_{(a,γa)}}↓ & & ↓^{f_{(b,γ_{b})}} \\ X_{(a,γ_{a})} & ⟵_{g_{(a,γa)(b,γ_{b})}} & X_{(b,g_{b})} \end{matrix}  (7)

commutes. Suppose that (a, γ_{b}^{1}), (a, γ_{b}^{2}), …, (a, γ_{b}^{n−1}) are defined for each a ∈ A with p(a) ≤ n−1 such that the each diagram (6) commutes. Let a ∈ A be a member of A with p(a) = n. This means that (a, γ_{b}^{1}), (a, γ_{b}^{2}), …, (a, γ_{b}^{n−1}) are defined. From the cofinitness of A it follows that the set of γ_{a}^{j} which are defined in Γ_{a} is finite. Hence there exists γ_{a}^{n} ≥ γ_{a}^{n−1}, …, γ_{a}^{1}. We define γ_{b}^{n} ∈ Γ_{b} considering the space X_{(a,γ_{a}^{n})} and the mappings f_{(a,γ_{a}^{n})}p_{ab}: X_{b} → X_{(a,γ_{a}^{n})}. Again, by Theorem 1 for each b ≥ a there exists a γ_{b}^{n} ∈ Γ_{b} such that for each γ_{b} ≥ γ_{b}^{n}, γ_{b}^{n−1}, …, γ_{b}^{1} and there is a mapping g_{(a,γ_{b})(b,γ_{b})} : X_{(b,γ_{b})} → X_{(a,γ_{a}^{n})} with f_{(a,γ_{a}^{n})}p_{ab} = g_{(a,γ_{b})(b,γ_{b})}f_{(b,γ_{b})}, i.e., the diagram

\begin{matrix}X_{a} & ⟵^{p_{ab}} & X_{b} \\ ^{f_{(a,γ^{n}_{a})}}↓ & & ↓^{f_{(b,γ_{b})}} \\ X_{(a,γ_{a}^{n})} & ⟵_{g_{(a,γ^{n}_{a})(b,γ_{b})}} & X_{(b,g_{b})} \end{matrix}  (8)

commutes. By induction on A (Lemma 7) the set D is defined. It remains to prove that {X_{(a,γ_{a})}, g_{(a,γ_{a})(b,γ_{b})}, A} is an approximate inverse system. Let U be a normal cover of X_{(a,γ_{a})}. Then V = f_{(a,γ_{a})}^{−1}(U) is a normal cover of X_{a}. By virtue of (A2) there exists a b ≥ a such that for each c ≥ d ≥ b we have (p_{ad}, p_{ca}p_{cd} ≤ V. By virtue of the commutativity of the diagrams of the form (8) it follows that

(g_{(a,γ_{a})(d,γ_{d})}, g_{(a,γ_{a})(c,γ_{c})}g_{(c,γ_{c})(d,γ_{d})}) ≤ V.

Thus, {X_{(a,γ_{a})}, g_{(a,γ_{a})(b,γ_{b})}, A} is an approximate inverse system.

Step 5. We define a partial order on D as follows. Let d_{1}, d_{2} be a pair of members of D such that d_{1} = {(a, γ_{a}) : a ∈ A, γ_{a} ∈ Γ_{a}} and d_{2} = {(a, δ_{a}) : a ∈ A, δ_{a} ∈ Γ_{a}}. We write d_{2} ≤ d_{1} if and only if δ_{a} ≤ γ_{a} for each a ∈ A. From Step 4. it follows that (D, ≤) is τ-directed. Moreover, X_{D} is a usual inverse system.

Step 6. For each d ∈ D the limit space X_{d} of the inverse system (5) is a compact space. Moreover, there exists a mapping F_{d}: X → X_{d}. The existence of F_{d} follows from the commutativity of the diagram (6). The following diagram illustrates the construction of d ∈ D and the space X_{d}.

\begin{matrix}X_{a} & ⟵^{p_{ab}} & X_{b} & ⟵^{p_{bc}} & X_{c} & ⟵^{p_{d}} & X \\ ↓^{f_{(a,δa)}} & ↓^{f_{(b,δ_{b})}} & ↓^{f_{(c,δc)}} \\ X_{(a,δ_{a})} & X_{(b,δ_{b})} & X_{(c,δ_{c})} \\ ↓^{f_{(a,γa)(a,δa)}} & ↓^{f_{(b,γ_{b})(b,δ_{b})}} & ↓^{f_{(c,γc)(c,δc)}} \\ X_{(a,γ_{a})} & ⟵_{g_{(a,γa)(b,γ_{b})}} & X_{(b,γ_{b})} & ⟵_{g_{(b,γ_{b})(c,γc)}} & X_{(c,γ_{c})} & ⟵_{g_{(c,γc)}} & X_{d} \\ ↓ & ↓ & ↓ \end{matrix}  (9)

Step 7. If d_{1}, d_{2} is a pair of members of D such that d_{1} = {(a, γ_{a}) : a ∈ A, γ_{a} ∈ Γ_{a}}, d_{2} = {(a, δ_{a}) : a ∈ A, δ_{a} ∈ Γ_{a}} and d_{2} ≥ d_{1}, then for each a ∈ A commutes the diagram

\begin{matrix}X_{(a,δ_{a})} & ⟵^{g_{(a,δa)(b,δ_{b})}} & X_{(b,δ_{b})} \\ ^{f_{(a,γa)(a,δa)}}↓ & & ↓^{f_{(b,γv)(b,δ_{b})}} \\ X_{(a,γ_{a})} & ⟵_{g_{(a,γa)(b,γ_{b})}} & X_{(b,γ_{b})} \end{matrix}  (10)

This follows from the surjectivity of the mappings f_{(b,γ_{b})} and from the commutativity of the diagrams of the form (6) for d_{1} and d_{2}, i.e., from the commutativity of the diagrams

\begin{matrix}X_{a} & ⟵^{p_{ab}} & X_{b} \\ ^{f_{(a,γa)}}↓ & & ↓^{f_{(b,γ_{b})}} \\ X_{(a,γ_{a})} & ⟵_{g_{(a,γa)(b,γ_{b})}} & X_{(b,g_{b})} \end{matrix}  (11)

and

\begin{matrix}X_{a} & ⟵^{p_{ab}} & X_{b} \\ ^{f_{(a,δa)}}↓ & & ↓^{f_{(b,δ_{b})}} \\ X_{(a,δ_{a})} & ⟵_{g_{(a,δa)(b,δ_{b})}} & X_{(b,d_{b})} \end{matrix}  (12)

Step 8. From Step 7. it follows that for d_{1}, d_{2} ∈ D with d_{2} ≥ d_{1} there exists a mapping F_{d_{1}d_{2}} : X_{d_{2}} → X_{d_{1}} (see [1, p. 138]) such that F_{d_{1}} = F_{d_{1}d_{2}}F_{d_{2}}.

Proof of Step 8. Let d_{1}, d_{2}, d_{3} ∈ D and let d_{1} ≤ d_{2} ≤ d_{3}. Then F_{d_{1}d_{3}} = F_{d_{1}d_{2}}F_{d_{2}d_{3}}. This follows from Step 7. and the commutativity condition in each inverse system X(a) = {X_{(a,γ)}, f_{(a,γ)(a,δ)}, Γ_{a}} (see (1) of Step 1.).

Step 9. The collection {X_{d}, F_{de}, D} is a usual inverse system of compact spaces.

Apply Steps 1.–8.

Step 10. There is a mapping F : X → X_{D} which is 1–1.

By Step 6. and Step 8. for each d ∈ D there is a mapping F_{d}: X → X_{d} such that F_{d_{1}} = F_{d_{1}d_{2}}F_{d_{2}} for d_{2} ≥ d_{1}. This means that there exists a mapping F: X → X_{D} [1, p. 138]. Let us prove that F is 1–1. Take a pair x, y of distinct points of X. There exists an a ∈ A such that x_{a} = p_{a}(x) and y_{a} = p_{a}(y) are distinct points of X_{a}. Now, there exists an (a, γ_{a}) such that f_{(a,γ_{a})}(x_{a}) and f_{(a,γ_{a})}(y_{a}) are distinct points of X_{(a,γ_{a})}. From Step 4. it follows that there is a d ∈ D such that F_{d}(x) and F_{d}(y) are distinct points of X_{d}. Thus, F is 1–1.

Step 11. The mapping F is a homeomorphism onto X_{D}. Let y be a point of X_{D}. Let us prove that there exists a point x ∈ X such that F (x) = y. For each d ∈ D we have a point y_{d} = F_{d}(y). Now, we have the points g_{(a,γ_{a})}F_{d}(y) in X_{(a,γ_{a})} and the subsets Y_{a} = f_{(a,γ_{a})}^{−1}(g_{(a,γ_{a})}F_{d}(y)) of X_{a}. Let U be an open neighborhood Y_{a}. There exists an open neighborhood V of g_{(a,γ_{a})}F_{d}(y) such that f_{(a,γ_{a})}^{−1}(V) ⊆ U. We infer that Ls{g_{(b,γ_{b})}(Y_{b}) : b ≥ a} ⊆ Y_{a} since g_{(a,γ_{a})}F_{d}(y) = lim{g_{(a,γ_{a})(b,γ_{b})}g_{(b,γ_{b})}F_{d}(y) : b ≥ a} and the diagrams (6) commute. By virtue of [6, Lemma 2.1] it follows that there exists a non-empty closed subset C_{d} of lim X such that p_{b}(C_{d}) ⊆ Y_{b}. The family {C_{d} : d ∈ D} has the finite intersection property. This means that X′ = ⋂{C_{d} : d ∈ D} is non-empty. For each x ∈ X′ we have F_{d}(x) = F_{d}(y), d ∈ D. Thus, F (y) = x. The proof is completed.

By the similar method of proof we obtain the following theorem.

Theorem 10. Let X = {X_{a}, p_{ab}, A} be an approximate inverse system of compact non-metric spaces with surjective bonding mappings p_{ab}. If each X_{a} is a limit of a usual τ-directed inverse system X(a) = {X_{(a,γ)}, f_{(a,γ)(a,δ)}, Γ_{a}} of compact spaces with w(X_{(a,γ)}) ≤ τ, then:

1. there exists a usual τ-directed inverse system X_{D} = {X_{d}, F_{de}, D} whose inverse limit X_{D} is homeomorphic to X = lim X,

2. every X_{d} is a limit of an approximate inverse system {X_{(a,γ_{a})}, g_{(a,γ_{a})(b,γ_{b})}, A} of compact spaces X_{(a,γ_{a})},

3. if the mappings p_{ab} and f_{(a,γ)(a,δ)} are monotone, then g_{(a,γ_{a})(b,γ_{b})} and F_{de} are monotone.

Corollary 3. Let X = {X_{n}, p_{nm}, ℕ} be an approximate inverse sequence of compact non-metric spaces with surjective bonding mappings p_{nm}. If each X_{n} is a limit of a usual σ-directed inverse system X(n) = {X_{(n,γ)}, f_{(n,γ)(n,δ)}, Γ_{n}} of metric compact spaces, then:

1. there exists a usual σ-directed inverse system X_{D} = {X_{d}, F_{de}, D} whose inverse limit X_{D} is homeomorphic to X = lim X,

2. every X_{d} is a limit of an approximate inverse sequence {X_{(n,γ_{n})}, g_{(n,γ_{n})(m,γ_{m})}, ℕ} of compact metric spaces X_{(n,γ_{n})},

3. if the mappings p_{nm} and f_{(n,γ)(n,δ)} are monotone, then g_{(n,γ_{n})(m,γ_{m})} and F_{de} are monotone. Let P be a topological property of spaces.

Theorem 11. Let X = {X_{a}, p_{ab}, A} be an approximate inverse system of compact non-metric spaces with surjective bonding mappings p_{ab} and let P be a topological property of spaces such that:

1. each X_{a} is a limit of a usual σ-directed inverse system X(a) = {X_{(a,γ)}, f_{(a,γ)(a,δ)}, Γ_{a}} of metric compact spaces with property P,

2. if X_{d} is a limit of an approximate inverse system {X_{(a,γ_{a})}, g_{(a,γ_{a})(b,γ_{b})}, A} of compact metric spaces X_{(a,γ_{a})} with property P, then X_{d} has P,

3. if Y is a limit of σ-directed usual inverse system of compact spaces with property P, then Y has P. Then X = lim X has the property P.

3. Applications

Lemma 9. Let X = {X_{n}, p_{nm}, ℕ} be an approximate inverse sequence of locally connected metric continua. If the bonding mappings are monotone and surjective, then X = lim X is locally connected.

Proof. There exists a usual inverse sequence Y = {Y_{i}, q_{ij}, M} such that Y_{i} = X_{n_{i}}, q_{ij} = p_{n_{i}n_{i+1}}p_{n_{i+1}n_{i+2}} … p_{n_{j−1}n_{j}} for each i, j ∈ ℕ and a homeomorphism H: lim X → lim Y [2, Proposition 8]. Each mapping q_{ij} as a composition of the monotone mappings is monotone. This means that Y is a usual inverse sequence of locally connected continua with monotone bonding mappings q_{ij}. Hence lim Y is locally connected. We infer that X = lim X is locally connected since there exists a homeomorphism H: lim X → lim Y.

Lemma 10. Let X = {X_{a}, p_{ab}, A} be an approximate inverse system of locally connected continua such that card(A) = ℵ_{0}. Then X = lim X is locally connected.

Proof. By virtue of Lemma 3 there exists a countable well-ordered subset B of A such that the collection {X_{b}, p_{bc}, B} is an approximate inverse sequence and lim X is homeomorphic to lim{X_{b}, p_{bc}, B}. From Lemma 9 it follows that lim{X_{b}, p_{bc}, B} is locally connected. Hence X = lim X is locally connected.

Lemma 11. Let X = {X_{a}, p_{ab}, A} be an approximate inverse system of locally connected metric continua and monotone bonding mappings. Then X = lim X is locally connected.

Proof. If card(A) = ℵ_{0} then we apply Lemma 10. If card(A) ≥ ℵ_{1} then from Corollary 1 it follows that X = lim X is homeomorphic to the limit of a σ-directed usual inverse system {X_{α}, q_{αβ}, Δ}, where each X_{α} is a limit of an approximate inverse subsystem {X_{γ}, p_{αβ}, Φ}, card(Φ) = ℵ_{0}. From Lemma 10 it follows that each X_{a} is locally connected. By Theorem 3 we infer that the limit of {X_{α}, q_{αβ}, Δ} is locally connected. Hence, X is locally connected since X is homeomorphic to lim{X_{α}, q_{αβ}, Δ}.

Theorem 12. Let X = {X_{a}, p_{ab}, A} be an approximate inverse system of locally connected continua and monotone bonding mappings. Then X = lim X is a locally connected continuum.

Proof. By virtue of Theorem 3 and Remark 1 every X_{a} is a limit of a usual σdirected inverse system X(a) = {X_{(a,γ)}, f_{(a,γ)(a,δ)}, Γ_{a}} of metric locally connected continua with monotone bonding mappings f_{(a,γ)(a,δ)}. From Theorem 9 it follows that there exist : 1) a usual σ-directed inverse system X_{D} = {X_{d}, F_{de}, D} whose inverse limit X_{D} is homeomorphic to X = lim X; 2) every X_{d} is a limit of an approximate inverse system {X_{(a,γ_{a})}, g_{(a,γ_{a})(b,γ_{b})}, A} of compact metric spaces X_{(a,γ_{a})} and 3) if the mappings p_{ab} and f_{(a,γ)(a,δ)} are monotone, then g_{(a,γ_{a})(b,γ_{b})} and F_{de} are monotone. Now, every X_{d} as the limit of approximate inverse system {X_{(a,γ_{a})}, g_{(a,γ_{a})(b,γ_{b})}, A} is locally connected because of Lemma 11. Finally, X is locally connected since X is homeomorphic to X_{D} = lim X_{D} and X_{D} is locally connected (Theorem 3).

We shall say that a non-empty compact space is perfect if it has no isolated points.

A continuum is said to be totally regular [12, p. 47] if for each x ≠ y in X there is a positive integer n and perfect subsets A_{1}, …, A_{n} of X such that x_{i} ∈ A_{i} for i = 1,…, n implies that {x_{1},…, x_{n}} separates x from y in X.

Lemma 12. [12, Proposition 7.4] Each totally regular continuum is hereditarily locally connected and rim-finite.

The following theorem is a part of [12, Theorem 7.15].

Theorem 13. If X is a continuum then the following conditions are equivalent:

1. X is totally regular,

2. X is homeomorphic to lim{X_{a}, f_{ab}, Γ} such that each X_{a} is a totally regular continuum and each f_{ab} is a monotone surjection.

Theorem 14. [12, Theorem 7.7] Let X = {X_{a}, p_{ab}, A} be an inverse system of totally regular continua X_{a} and monotone surjective mappings p_{ab}. Then X = lim X is totally regular.

Theorem 15. Let X be a non-metric totally regular continuum. There exists a σ-directed inverse system X = {X_{a}, p_{ab}, A} such that each X_{a} is totally regular, each f_{ab} is a monotone surjection and X is homeomorphic to lim X.

Proof. Apply [12, Theorem 9.4], Theorem 14 and Lemma 3.5 of [14].

Now we shall prove the following theorem.

Theorem 16. Let X = {X_{n}, p_{nm}, ℕ} be an approximate inverse sequence of totally regular metric continua. If the bonding mappings are monotone and surjective, then X = lim X is totally regular.

Proof. There exists a usual inverse sequence Y = {Y_{i}, q_{ij}, M} such that Y_{i} = X_{n_{i}}, q_{ij} = p_{n_{i}n_{i+1}}p_{n_{i+1}n_{i+2}} … p_{n_{j−1}n_{j}} for each i, j ∈ ℕ and a homeomorphism H: lim X → lim Y [2, Proposition 8]. Each mapping q_{ij} as a composition of the monotone mappings is monotone. This means that Y is a usual inverse sequence of totally regular continua with monotone bonding mappings q_{ij}. By virtue of Theorem 14 lim Y is totally regular. We infer that X = lim X is totally regular since there exists a homeomorphism H: lim X → lim Y.

Theorem 17. Let X = {X_{a}, p_{ab}, A} be an approximate inverse system of totally regular continua such that card(A) = ℵ_{0}. Then X = lim X is totally regular.

Proof. By virtue of Lemma 3 there exists a countable well-ordered subset B of A such that the collection {X_{b}, p_{bc}, B} is an approximate inverse sequence and lim X is homeomorphic to lim{X_{b}, p_{bc}, B}. From Theorem 16 it follows that lim{X_{b}, p_{bc}, B} is totally regular. Hence X = lim X is totally regular.

Theorem 18. Let X = {X_{a}, p_{ab}, A} be an approximate inverse system of totally regular continua and monotone bonding mappings. If w(X_{a}) < τ < card(A) for each a ∈ A, then X = lim X is a totally regular continuum.

Proof. By virtue of Theorem 7 (for λ = ℵ_{0}) there exists a σ-directed inverse system {X_{α}, q_{αβ}, T }, where each X_{α} is a limit of an approximate inverse subsystem {X_{γ}, p_{αβ}, Φ}, card(Φ) = ℵ_{0}. From Theorem 17 it follows that every X_{α} is totally regular. Theorem 14 completes the proof.

Theorem 19. Let X = {X_{a}, p_{ab}, A} be an approximate inverse system of totally regular metric continua and monotone bonding mappings. Then X = lim X is totally regular continuum.

Proof. If card(A) = ℵ_{0} then we apply Theorem 17. If card(A) ≥ ℵ_{1} then from Theorem 18 it follows that X is totally regular.

Theorem 20. Let X = {X_{a}, p_{ab}, A} be an approximate inverse system of totally regular non-metric continua with surjective monotone bonding mappings p_{ab}. Then X = lim X is totally regular.

Proof. By virtue of Theorem 15 every X_{a} is a limit of a usual σ-directed inverse system X(a) = {X_{(a,γ)}, f_{(a,γ)(a,δ)}, Γ_{a}} of metric totally regular continua with monotone bonding mappings f_{(a,γ)(a,δ)}. From Theorem 9 it follows that there exist: 1) a usual σ-directed inverse system X_{D} = {X_{d}, F_{de}, D} whose inverse limit X_{D} is homeomorphic to X = lim X, 2) every X_{d} is a limit of an approximate inverse system {X_{(a,γ_{a})}, g_{(a,γ_{a})(b,γ_{b})}, A} of compact metric spaces X_{(a,γ_{a})} and 3) if the mappings p_{ab} and f_{(a,γ)(a,δ)} are monotone, then g_{(a,γ_{a})(b,γ_{b})} and F_{de} are monotone. Now, every X_{d} as the limit of approximate inverse system {X_{(a,γ_{a})}, g_{(a,γ_{a})(b,γ_{b})}, A} is totally regular because of Theorem 19. Finally, X is totally regular since X is homeomorphic to X_{D} = lim X_{D} and X_{D} is totally regular (Theorem 14).

Theorem 21. Let X = {X_{a}, p_{ab}, A} be an approximate inverse system of totally regular continua with surjective monotone bonding mappings p_{ab}. Then X = lim X is totally regular.

Proof. Apply Theorems 19 and 20.

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(received 08.05.2000, in revised form 23.12.2000)

Faculty of Organization and Informatics Varaždin, Croatia E-mail: ivan.loncar1@vz.tel.hr