The Locally Finite Topology on 2 X

Beer, Gerald; Himmelberg, C. J.; Prikry, K.; Vleck, F. S. Van

doi:10.2307/2046570

Cited by 18 publications

(20 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This statement also holds if D2 is replaced by Di or Dq. Consequently, the various Hausdorff metric uniformities on 2X, one corresponding to each d G Di, together form a subbase for 2^(T\ Thus, we have the following result, proved in [4]: THEOREM 2.6. Let (X,t) be a metrizable space.…”

Section: ^«|mentioning

confidence: 87%

“…Theorems 12.18 and 15.1 of [12]), so that we have the following characterization of UC spaces given in [4]. COROLLARY 2.5.…”

Section: ^«|mentioning

confidence: 98%

“…In a recent paper [4], Beer et al proved that the locally finite topology eT on the hyperspace 2X of a metrizable space X is the supremum of all the Hausdorff metric topologies corresponding to equivalent metrics on X. In view of this result, and because a Hausdorff metric topology generally lacks topological invariance, the authors of [4] justifiably contended that the locally finite topology on a hyperspace would be a more useful structure in most applications.…”

Section: Introductionmentioning

confidence: 99%

“…In view of this result, and because a Hausdorff metric topology generally lacks topological invariance, the authors of [4] justifiably contended that the locally finite topology on a hyperspace would be a more useful structure in most applications.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Fine uniformity and the locally finite hyperspace topology

Naimpally¹,

Sharma²

1988

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

ABSTRACT. It is shown that a uniformizable space X is normal iff the locally finite topology eT on the hyperspace 2X coincides with the topology transmitted by the fine uniformity of X. We also prove that, for X normal, the topology eT is first countable only if the set of limit points X' of X is countably compact. Applications of these results to pseudocompactness and Atsuji spaces are given. Introduction.In a recent paper [4], Beer et al. proved that the locally finite topology eT on the hyperspace 2X of a metrizable space X is the supremum of all the Hausdorff metric topologies corresponding to equivalent metrics on X. In view of this result, and because a Hausdorff metric topology generally lacks topological invariance, the authors of [4] justifiably contended that the locally finite topology on a hyperspace would be a more useful structure in most applications.In this paper, we determine the appropriate context for the locally finite topology, namely that this topology is a uniform notion. Specifically, we prove that, in the class of uniformizable spaces, it is precisely the normal spaces for which the locally finite topology on a hyperspace coincides with the topology transmitted by the fine uniformity of the space. This result specialized to metrizable spaces yields the aforementioned result of Beer et al. We also show that if X is paracompact, then for eT to be first countable, X' must be compact. From this, several known characterizations of Atsuji spaces follow. All topological spaces considered in this paper are assumed to be Ti. Much of our terminology about hyperspaces is the same as in Michael's paper [11].Let (X, t) be a Ti space. The hyperspace 2X oí X is the set {E Ç X: E is closed, E ^ 0}. For a collection sf of subsets of X, we write

show abstract

Section: ^«|mentioning

confidence: 87%

“…Theorems 12.18 and 15.1 of [12]), so that we have the following characterization of UC spaces given in [4]. COROLLARY 2.5.…”

Section: ^«|mentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Fine uniformity and the locally finite hyperspace topology

Naimpally¹,

Sharma²

1988

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…(a) X is normal if and only if the H-B uniformity associated with the fine uniformity induces the locally finite topology ( [14,20]). (b) In a metrisable space, the supremum of all Hausdorff metric topologies, associated with compatible metrics, is the locally finite topology (see [20] for a non-standard treatment and [2] for the usual one).…”

Section: Proof It Is Sufficient To Show That If a ∈ Ementioning

confidence: 99%

All hypertopologies are hit-and-miss

Naimpally

2002

Appl. Gen. Topol.

View full text Add to dashboard Cite

Abstract. We solve a long standing problem by showing that all known hypertopologies are hit-and-miss. Our solution is not merely of theoretical importance. This representation is useful in the study of comparison of the Hausdorff-Bourbaki or H-B uniform topologies and the Wijsman topologies among themselves and with others. Up to now some of these comparisons needed intricate manipulations. The H-B uniform topologies were the subject of intense activity in the 1960's in connection with the Isbell-Smith problem. We show that they are proximally locally finite topologies from which the solution to the above problem follows easily. It is known that the Wijsman topology on the hyperspace is the proximal ball (hit-and-miss) topology in"nice" metric spaces including the normed linear spaces. With the introduction of a new far-miss topology we show that the Wijsman topology is hit-andmiss for all metric spaces. From this follows a natural generalization of the Wijsman topology to the hyperspace of any T1 space. Several existing results in the literature are easy consequences of our work.2000 AMS Classification: 54B20, 54E05, 54E15.

show abstract

On the Fell topology

Beer

1993

Set-Valued Anal

View full text Add to dashboard Cite

show abstract

The Locally Finite Topology on 2 X

Cited by 18 publications

References 0 publications

Fine uniformity and the locally finite hyperspace topology

Fine uniformity and the locally finite hyperspace topology

All hypertopologies are hit-and-miss

On the Fell topology

Contact Info

Product

Resources

About