Proceedings of the Ninth Prague Topological Symposium Contributed papers from the symposium held in Prague, Czech Republic, August 19–25, 2001

FELL-CONTINUOUS SELECTIONS AND TOPOLOGICALLY WELL-ORDERABLE SPACES II

VALENTIN GUTEV

Abstract. The present paper improves a result of [3] by showing that a space X is topologically well-orderable if and only if there exists a selection for F_{2}(X) which is continuous with respect to the Fell topology on F_{2}(X). In particular, this implies that F(X) has a Fell-continuous selection if and only if F_{2}(X) has a Fell-continuous selection.

1. Introduction

Let X be a topological space, and let F(X) be the family of all non-empty closed subsets of X. Also, let τ be a topology on F(X) and D ⊂ F(X). A map f : D → X is a selection for D if f(S) ∈ S for every S ∈ D. A map f : D → X is a τ-continuous selection for D if it is a selection for D which is continuous with respect to the relative topology τ on D as a subspace of F(X).

Two topologies on F(X) will play the most important role in this paper. The first one is the Vietoris topology τ_{V} which is generated by all collections of the form

⟨V⟩ = {S ∈ F(X) : S ∩ V ≠ ∅, V ∈ V, and S ⊂ ⋃ V} ,

where V runs over the finite families of open subsets of X. The other one is the Fell topology τ_{F} which is defined by all basic Vietoris neighbourhood ⟨V⟩ with the property that X \ ⋃ V is compact.

Finally, let us recall that a space X is topologically well-orderable (see Engelking, Heath and Michael [2]) if there exists a linear order “≺” on X such that X is a linear ordered topological space with respect to ≺, and every non-empty closed subset of X has a ≺-minimal element.

2000 Mathematics Subject Classification. Primary 54B20, 54C65; Secondary 54D45, 54F05.

Key words and phrases. Hyperspace topology, selection, ordered space, local compactness.

Valentin Gutev, Fell-continuous selections and topologically well-orderable spaces II, Proceedings of the Ninth Prague Topological Symposium, (Prague, 2001), pp. 147–153, Topology Atlas, Toronto, 2002; arXiv:math.GN/0204129.

Recently, the topologically well-orderable spaces were characterized in [3, Theorem 1.3] by means of Fell-continuous selections for their hyperspaces of non-empty closed subsets.

Theorem 1.1 ([3]). A Hausdorff space X is topologically well-orderable if and only if F(X) has a τ_{F} -continuous selection.

In the present paper, we improve Theorem 1.1 by showing that one may use τ_{F} -continuous selections only for the subset F_{2}(X) = {S ∈ F(X) : |S| ≤ 2} of F(X). Namely, the following theorem will be proven.

Theorem 1.2. A Hausdorff space X is topologically well-orderable if and only if F_{2}(X) has a τ_{F} -continuous selection.

About related results for Vietoris-continuous selections, the interested reader is referred to van Mill and Wattel [6].

Theorem 1.2 is interesting also from another point of view. According to Theorem 1.1, it implies the following result which may have an independent interest.

Corollary 1.3. If X is a Hausdorff space, then F(X) has a τ_{F} -continuous selection if and only if F_{2}(X) has a τ_{F} -continuous selection.

A word should be said also about the proof of Theorem 1.2. In general, it is based on the proof of Theorem 1.1 stated in [3], and is separated in a few different steps which are natural generalizations of the corresponding ones given in [3]. In fact, the paper demonstrates that all statements of [3] remain true if F(X) is replaced by F_{2}(X). Related to this, the interested reader may consult an alternative proof of Theorem 1.2 given in [1] and based again on the scheme in [3].

2. A reduction to locally compact spaces

In the sequel, all spaces are assumed to be at least Hausdorff.

In this section, we prove the following generalization of [3, Theorem 2.1].

Theorem 2.1. Let X be a space such that F_{2}(X) has a τ_{F} -continuous selection. Then X is locally compact.

Proof. We follow the proof of [3, Theorem 2.1]. Namely, let f be a τ_{F} - continuous selection for F_{2}(X) and suppose, if possible, that X is not locally compact. Hence, there exists a point p ∈ X such that \overline{V} is not compact for every neighbourhood V of p in X. Claim that there exists a point q ∈ X such that

(1) q ≠ p and f({p, q}) = p.

To this purpose, note that there exists F ∈ F(X) such that F is not compact and p ∉ F. Then, f^{−1}(X \ F) is a τ_{F} -neighbourhood of {p} in F_{2}(X), so there exists a finite family W of open subsets of X such that X \ ⋃ W is compact and

{p} ∈ ⟨W⟩ ∩ F_{2}(X) ⊂ f^{−1}(X \ F).

Then, F ∩ W ≠ ∅ for some W ∈ W because F is not compact. Therefore, there exists a point q ∈ F ∩ (⋃ W). This q is as required.

Let q be as in (1). Since X is Hausdorff, f({q}) ≠ f({p, q}), and f is τ_{F} -continuous, there now exist two finite families U and V of open subsets of X such that X \ ⋃ U is compact, {q} ∈ ⟨U⟩, {p, q} ∈ ⟨V⟩, and ⟨U⟩ ∩ ⟨V⟩ = ∅. Then,

(2) p ∈ V_{p} = ⋂{V ∈ V : p ∈ V } ⊂ X \ ⋃ U.

Indeed, suppose there is a point ℓ ∈ V_{p} ∩ (⋃ U). Then, {ℓ, q} ∈ ⟨U⟩ because {q} ∈ ⟨U⟩. However, we also get that {ℓ, q} ∈ ⟨V⟩ because q ∉ V for some V ∈ V implies p ∈ V, hence ℓ ∈ V_{p} ⊂ V. Thus, we finally get that {ℓ, q} ∈ ⟨U⟩ ∩ ⟨V⟩ which is impossible. So, (2) holds as well.

To finish the proof, it remains to observe that this contradicts the choice of p. Namely V_{p} becomes a neighbourhood of p which, by (2), has a compact closure because X \ ⋃ U is compact. □

3. A reduction to compact spaces

For a locally compact space X we will use αX to denote the one point compactification of X. For a non-compact locally compact X let us agree to denote by α the point of the singleton αX \ X.

In what follows, to every family D ⊂ F(X) we associate a family α(D) ⊂ F(αX) defined by

α(D) = {S ∈ F(αX) : S ∩ X ∈ D ∪ {∅}}.

The following extension theorem was actually proven in [3, Theorem 3.1].

Theorem 3.1. Let X be a locally compact non-compact space X, and D ⊂ F(X). Then, D has a τ_{F} -continuous selection if and only if α(D) has a τ_{V} -continuous selection g such that g^{−1}(α) = {{α}}.

Proof. Just the same proof as in [3, Theorem 3.1] works. Namely, if f is a τ_{F} -continuous selection for D, we may define a selection g for α(D) by g(S) = f(S ∩ X) if S ∩ X ≠ ∅ and g(S) = α otherwise, where S ∈ α(D). Clearly g^{−1}(α) = {{α}} and, as shown in [3, Theorem 3.1], g is τ_{V} -continuous. If now g is a τ_{V} -continuous selection for α(D), with g^{−1}(α) = {{α}}, then g(S ∪ {α}) ∈ S for every S ∈ D, so we may define a selection f for D by f(S) = g (S ∪ {α}), S ∈ D. The verification that f is τ_{F} -continuous was done in [3, Theorem 3.1]. □

4. Special selections and connected sets

In what follows, to every selection f : F_{2}(X) → X we associate an orderlike relation “≺_{f}” on X (see Michael [5]) defined for x ≠ y by

x_{1} ≺_{f} x_{2} iff f({x_{1}, x_{2}}) = x_{1}.

Further, we will need also the following ≺_{f}-intervals:

(x, +∞)_{≺_{f}} = {z ∈ X : x ≺_{f} z}

and

[x, +∞)_{≺_{f}} = {z ∈ X : x ⪯_{f} z}.

Now, we provide the generalization of [3, Theorem 4.1] for the case of F_{2}(X).

Theorem 4.1. Let X be a space, a ∈ X, and let A ∈ F(X) be a connected set such that |A| > 1 and a ∈ A ∩ \overline{X \ A}. Also, let f : F_{2}(X) → X be a τ_{V} -continuous selection for F_{2}(X). Then, f^{−1}(a) ≠ {{a}}.

Proof. Suppose, on the contrary, that f^{−1}(a) = {{a}}. By hypothesis, there exists a point b ∈ A, with b ≠ a. Since f is τ_{V} -continuous, f({a, b}) = b and a ∈ \overline{X \ A}, we can find a point c ∈ X \ A such that f(\{b, c\}) = b. Then, B = A ∩ (c, +∞)_{≺_{f}} is a clopen subset of A because B = A ∩ [c, +∞)_{≺_{f}}, see [5]. However, this is impossible because b ∈ A \ B, while a ∈ B. □

5. A further result about special selections

Following [3], we shall say that a point a ∈ X is a partition of X if there are open subset L, R ⊂ X \ \{a\} such that \overline{L} ∩ \overline{R} = \{a\} and L ∩ R = ∅.

We finalize the preparation for the proof of Theorem 1.2 with the following result about special Vietoris continuous selections and partitions which generalizes [3, Theorem 5.1].

Theorem 5.1. Let X be a compact space, f a τ_{V} -continuous selection for F_{2}(X), and let a ∈ X be a partition of X such that f^{−1}(a) = {{a}}. Then, X is first countable at a.

Proof. By definition, there are open sets L, R ⊂ X \ \{a\} such that \overline{L} ∩ \overline{R} = {a} and L ∩ R = ∅. Hence, both L and R are non-empty. Take a point ℓ_{0} ∈ L. Then, by hypothesis, f({ℓ_{0}, a}) = ℓ_{0}. Since f is τ_{V} -continuous, this implies the existence of a neighbourhood L_{0} ⊂ L of ℓ_{0} and a neighbourhood V_{0} of a such that

L_{0} ∩ V_{0} = ∅ and f(⟨{L_{0}, V_{0}}⟩ ∩ F_{2}(X)) ⊂ L_{0}.

Since a ∈ \overline{R}, there exists a point r_{0} ∈ V_{0} ∩ R. Observe that f(\{a, r_{0}\}) = r_{0} ∈ V_{0}. Hence, just like before, we may find a neighbourhood R_{0} ⊂ R ∩ V_{0} of r_{0} and a neighbourhood W_{0} ⊂ V_{0} of a such that

R_{0} ∩ W_{0} = ∅ and f(⟨{R_{0}, W_{0}}⟩ ∩ F_{2}(X)) ⊂ R_{0}.

Thus, by induction, we may construct a sequence {ℓ_{n} : n < ω} of points of L, a sequence {r_{n} : n < ω} of points of R, and open sets L_{n}, V_{n}, R_{n}, W_{n} ⊂ X such that

(3)  \begin{matrix}ℓ_{n} ∈ L_{n}, \\ a ∈ V_{n}, \\ L_{n} ∩ V_{n} = ∅ and \\ f(⟨\{L_{n}, V_{n}\}⟩ ∩ F_{2}(X)) ⊂ L_{n}, \end{matrix}

(4)  \begin{matrix}r_{n} ∈ R_{n}, \\ a ∈ W_{n}, \\ R_{n} ∩ W_{n} = ∅ and \\ f(⟨\{R_{n}, W_{n}\}⟩ ∩ F_{2}(X)) ⊂ R_{n}, \end{matrix}

and

(5)  \begin{matrix}V_{n+1} ⊂ W_{n} ⊂ V_{n}, \\ L_{n+1} ⊂ L ∩ W_{n} and \\ R_{n} ⊂ R ∩ V_{n}. \end{matrix}

Since X is compact, {ℓ_{n} : n < ω} has a cluster point ℓ, and {r_{n} : n < ω} has a cluster point r. We claim that ℓ = r. Indeed, suppose for instance that ℓ ≺_{f} r (the case r ≺_{f} ℓ is symmetric). Then, there are disjoint open sets U_{ℓ} and U_{r} such that ℓ ∈ U_{ℓ}, r ∈ U_{r}, and x ≺_{f} y for every x ∈ U_{ℓ} and y ∈ U_{r}, see [4]. Next, take ℓ_{n} ∈ U_{ℓ} and r_{m} ∈ U_{r} such that n > m. Then, we have ℓ_{n} ≺_{f} r_{m}. However, by (3), (4) and (5), we get that {r_{m}, ℓ_{n}} ∈ ⟨{R_{m}, W_{m}}⟩ ∩ F_{2}(X), and therefore f({r_{m}, ℓ_{n}}) = r_{m}. This is clearly impossible, so ℓ = r.

Having already established this, let us observe that b = ℓ = r implies b ∈ \overline{L} ∩ \overline{R} because ℓ ∈ \overline{L} and r ∈ \overline{R}. However, \overline{L} ∩ \overline{R} = \{a\} which finally implies that b = a.

We are now ready to prove that, for instance, {W_{n} : n < ω} is a local base at a. To this end, suppose if possible that this fails. Hence, there exists an open neighbourhood U of a such that W_{n} \ U ≠ ∅ for every n < ω. Next, whenever n < ω, take a point t_{n} ∈ W_{n} \ U. Since X is compact, {t_{n} : n < ω} has a cluster point t ∉ U. Then, t ≺_{f} a and, as before, we may find disjoint open sets U_{t} and U_{a} such that t ∈ U_{t}, a ∈ U_{a}, and x ≺_{f} y for every x ∈ U_{t} and y ∈ U_{a}. Next, take t_{n} ∈ U_{t} and r_{m} ∈ U_{a} such that n > m. Then, t_{n} ≺_{f} r_{m}, while, by (4) and (5), r_{m} ≺_{f} t_{n} because {r_{m}, t_{n}} ∈ ⟨{R_{m}, W_{m}}⟩ ∩ F_{2}(X). The contradiction so obtained completes the proof. □

6. Proof of Theorem 1.2

In case X is a topologically well-orderable space, we may use Theorem 1.1.

Suppose that F_{2}(X) has a τ_{F} -continuous selection. If X is compact, then Theorem 1.2 is, in fact, a result of van Mill and Wattel [6]. Let X be noncompact. By Theorem 2.1, X is locally compact. Then, by Theorem 3.1, F_{2}(αX) has a τ_{V} -continuous selection f such that f^{−1}(α) = {{α}}. Relying once again on the result of [6], αX is a linear ordered topological space with respect to some linear order “<” on αX. It now suffices to show that there exists a compatible (with the topology of αX) linear order “≺” on αX such that α is either the first or the last element of αX, see [2, Lemma 4.1]. We show this following precisely the proof of Theorem 1.1 in [3]. Namely, let

L = {x ∈ αX : x < α} and R = {x ∈ αX : α < x}.

Note that L, R ⊂ αX \ {α} = X are open subsets of αX. In case one of these sets is also closed, the desired linear order “≺” on αX can be defined by exchanging the places of L and R. Namely, by letting for x, y ∈ αX that x ≺ y if and only if

x, y ∈ \overline{L} and x < y, or
x, y ∈ \overline{R} and x < y, or
x ∈ \overline{R} and y ∈ \overline{L}.

Finally, let us consider the case \overline{L} ∩ \overline{R} = \{α\}. Then, α is a partition of αX. Hence, by Theorem 5.1, αX is first countable at α. Let C[α] be the connected component of α in αX. Since f^{−1}(α) = {{α}}, it now follows from Theorem 4.1 that C[α] = {α}. Indeed, C′ = C[α] ∩ {x ∈ αX : x ≤ α} and C′′ = C[α] ∩ {x ∈ αX : x ≥ α} are both connected subsets of X with α ∈ C′ ∩ \overline{(X \ C′)} and α ∈ C′′ ∩ \overline{(X \ C′′)} (consider that X \ C′ ⊃ R and X \ C′′ ⊃ L), so that C′ = {α} and C′′ = {α}, whence also C[α] = {α}. Then, αX has a clopen base at α. Indeed, let ℓ ∈ L and r ∈ R. Since C[α] is also the quasi-component of the point α, there are clopen neighbourhoods U_{ℓ}, U_{r} of α such that ℓ ∉ U_{ℓ} and r ∉ U_{r}. Then,

U = {x ∈ U_{ℓ} ∩ U_{r} : ℓ < x < r} = {x ∈ U_{ℓ} ∩ U_{r} : ℓ ≤ x ≤ r}

is a clopen neighbourhood of α with U ⊂ {x ∈ X : ℓ < x < r}.

That is, αX has a clopen base at α and it is first countable at this point. Then, let {U_{n} : n < ω} be a decreasing clopen base at α, with U_{0} = αX. Next, for every point x ∈ X, let n(x) = max{n : x ∈ U_{n}} and, for convenience, n(α) = ω. Finally, we may define a linear order “≺” on αX by putting x ≺ y if and only if

either n(x) < n(y) or n(x) = n(y) and x < y.

Since {U_{n} : n < ω} is a decreasing clopen base at α, the order “≺” is compatible with the topology of αX. It is clear that, with respect to “≺”, α is the last element of X. This completes the proof.

References

[1] Giuliano Artico and Umberto Marconi, Selections and topologically well-ordered spaces, Topology Appl. 115 (2001), no. 3, 299–303. MR 1 848 130

[2] R. Engelking, R. W. Heath, and E. Michael, Topological well-ordering and continuous selections, Invent. Math. 6 (1968), 150–158. MR 39 #6272 [3] V. Gutev and T. Nogura, Fell continuous selections and topologically well-orderable

spaces, Internal Report 13/99, University of Natal, 1999.

[4] , Selections and order-like relations, Applied General Topology 2 (2001), no. 2, 205–218.

[5] E. Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152– 182. MR 13,54f

[6] Jan van Mill and Evert Wattel, Selections and orderability, Proc. Amer. Math. Soc. 83 (1981), no. 3, 601–605. MR 82i:54038

School of Mathematical and Statistical Sciences, Faculty of Science, University of Natal, King George V Avenue, Durban 4041, South Africa

   E-mail address: gutev@nu.ac.za