Selections for hyperspace topologies

Valentin Gutev

School of Mathematical and Statistical Sciences,
Faculty of Science, University of Natal,
King George V Avenue,
Durban 4041, South Africa
gutev@sci.und.ac.za

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1 The concept of a t-continuous selection

Let X be a T₁-space, F(X) be the set of all non-empty closed subsets of X, and let t be an admissible topology on F(X) in the sense that the restriction of t on {{x} : x in X} coincides with the topology of X.

Let D be a subset of F(X). A map f:D --> X is a selection for D if f(S) is in S for every S in D. A map f:D --> X is a t-continuous selection for D if it is a selection for D which is continuous with respect to the relative topology on D as a subspace of (F(X),t).

So far, one of the best known admissible topologies on F(X) is the Vietoris one t_V. Let us recall that t_V is generated by all collections of the form

<V> = { S in F(X) : S is a subset of the union of V and S meets V in a non-empty intersection for each V in V },

2 Selections and orderability

In what follows, all spaces are assumed to be at least Hausdorff. For a space X, let F₂(X) = {S in F(X): |S| <= 2}. Also, let Sel(X) be the set of all t_V-continuous selections for F(X), and Sel₂(X) that of all t_V-continuous selections for F₂(X).

Any selection f in Sel₂(X) naturally defines an order-like relation <_f on X [9] by letting for x no equal to y that x <_f y iff f({x,y}) = x. Unfortunately, in general, <_f fails to be a linear order on X. Let us denote by T_f the topology generated by all possible ``open'' <_f-intervals.

Theorem 1 [9] If (X,T) is a topological space and f in Sel₂(X), then T_f is a subset T. If, in addition, (X,T) is connected, then <_f is a proper linear order on X.

Theorem 2 [11] If X is a compact space with Sel₂(X) non-empty, then it is a linear ordered topological space. In particular, ind(X) <= 1.

Here, ind(X) means the small inductive dimension of X.

In view of Theorem 2.1, it makes some sense to investigate the topology T_f. In particular, Theorems 2.1 and 2.2 might be associated by the following natural question.

Question 1. Let (X,T) be a locally compact space with Sel₂(X) non-empty. Does there exist a topology T_* subset T on X such that(X,T_*) is a linear ordered topological space?

Here is a related question.

Question 2. Does there exist a space X such that Sel₂(X) non-empty and ind(X) > 1?

3 On the cardinality of Sel(X)

Theorem 1 For a space X, with Sel(X) non-empty, the following holds:

1. If X is connected, then |Sel(X)| <= 2, [9].
2. Sel(X) is finite iff X has finitely many connected components, [13].
3. If X is infinite and connected, then |Sel(X)| = 2 iff X is compact, [12].

For some other relations between |Sel(X)| and X, the interested reader is referred to [5, 12, 13].

4 On the variety of Sel(X)

Theorem 1 [8] Let X be a space with Sel(X) non-empty. Then, {f(X) : f in Sel(X)} is dense in X provided X is zero-dimensional, while X is totally disconnected provided {f(X) : f in Sel(X)} is dense in X.

As usual, a space X is zero-dimensional if it has a base of clopen sets, i.e. if ind(X) = 0.

Question 3. Does there exist a space X which is not zero-dimensional but {f(X) : f in Sel(X)} is dense in X?

Question 4. Let X be a totally disconnected space with Sel(X) non-empty. Is the set {f(X) : f in Sel(X)} dense in X?

Some related results are summarized below.

Theorem 2 For a space X, with Sel(X) non-empty, the following holds:

1. X is zero-dimensional provided for every x in X there exists f_x in Sel(X) with f_x^-1(x) = {S in F(X) : x in S}, [8].
2. If X is first countable and zero-dimensional, then for every x in X there exists f_x in Sel(X) with f_x^-1(x) = {S in F(X) : x in S}, [8].
3. If X is separable, then it is zero-dimensional and first countable iff for every x in X there exists f_x in Sel(X) with f_x^-1(x) = {S in F(X) : x in S}, [5].

5 Selections in metrizable spaces

Here, dim(X) means the covering dimension of X.

Most of the hypotheses in Theorem 5.1 are the best possible. The assumption dim(X) = 0 cannot be dropped or even weakened to dim(X) <= 1 [4, 11]; a metrizable space X is completely metrizable provided there exists a t_V-continuous selection for F(X) [10]; the continuity of f can be improved in several directions involving hyperspace topologies weaker than the Vietoris one [1, 3, 6, 7]. For some open questions on the continuity of f, the interested reader is referred to [3, 7].

Concerning the hypothesis "dim(X) = 0", the following question seems to be interesting.

Question 5. Does there exist a zero-dimensional metrizable space X such that F(X) has a t_V-continuous selection but dim(X) is not equal to 0?

References

[1]

D. Bertacchi and C. Costantini, Existence of selections and disconnectedness properties for the hyperspace of an ultrametric space, Topology Appl. 88 (1998), 179-197.

[2]

M. Choban, Many-valued mappings and Borel sets. I, Trans. Moscow Math. Soc. 22 (1970), 258-280.

[3]

C. Costantini and V. Gutev, Recognizing special metrics by topological properties of the ``metric''-proximal hyperspace, preprint, 1997.

[4]

R. Engelking, R. W. Heath, and E. Michael, Topological well-ordering and continuous selections, Invent. Math. 6 (1968), 150-158.

[5]

S. García-Ferreira, V. Gutev, T. Nogura, M. Sanchis, and A. Tomita, Extreme selections for hyperspaces of topological spaces, Topology Appl. (2000), to appear.

[6]

V. Gutev, Selections and hyperspace topologies via special metrics, Topology Appl. 70 (1996), 147-153.

[7]

V. Gutev and T. Nogura, Selections for Vietoris-like hyperspace topologies, Proc. London Math. Soc. 80 (2000), no. 3, 235-256.

[8]

----, Vietoris continuous selections and disconnectedness-like properties, Proc. Amer. Math. Soc. (2000), to appear.

[9]

E. Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152-182.

[10]

J. van Mill, J. Pelant, and R. Pol, Selections that characterize topological completeness, Fund. Math. 149 (1996), 127-141.

[11]

J. van Mill and E. Wattel, Selections and orderability, Proc. Amer. Math. Soc. 83 (1981), no. 3, 601-605.

[12]

T. Nogura and D. Shakhmatov, Characterizations of intervals via continuous selections, Rendiconti del Circolo Matematico di Palermo, Serie II, 46 (1997), 317-328.

[13]

----, Spaces which have finitely many continuous selections, Bollettino dell'Unione Matematica Italiana 11-A (1997), no. 7, 723-729.

Date published: September 15, 2000.