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On Wikipedia I found the notion of Kuratowski convergence. There is stated, that this convergence is equivalent to the convergence with respect to the Hausdorff metric on sets, if the ambient space is a compact metric space. As reference there is given the book "Topology" by Kuratowski. I went to the library and started reading. At first I did not find this notion of set convergence, but by this post on MSE I noticed I read in the wrong part of the book. As it is answered in this post in the book there is only stated the definition of the Kuratowski limit (as far as I see) and the statement that for closed sets in a bounded ambient space the convergence in the Hausdorff metric implies the Kuratowski convergence. Concerning the assertion stated by Wikipedia about compact ambient spaces I found this post on MSE whose answers provides also a proof for the other direction of the equivalence if the ambient space is a compact metric space. So where does this statement come from if not from the book mentioned above?

What I wish for is an additional reference for this equivalence statement (I would be also happy if you prove me wrong and the statement is contained in that book). I suppose after 50 years there should be another book including the Kuratowski convergence and related statements.

EDIT: It seems that I have missed this post which actually asks nearly the same question as I. The book given there does not give a further reference on the provided statements, not even for the definition of Kuratowski convergence. This is not a problem, but if there is another book or paper you know I would appreciate any answer mentioning it.

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It is quite well documented in "Topologies on Closed and Closed Convex sets" by Gerald Beer.

On page 143 Corollary 5.1.11 it is shown that the Haussdorf metric topology is equivalent to the Fell topology on the Closed non-empty subsets of a compact metric space and on page 147 it is mentioned with reference that the Fell convergence is equivalent with Kuratowski-Painleve convergence (Kuratowski convergence) On a Cech-complete Haussdorf space. (A compact metric space is a Cech-complete Haussdorf space.)

G. Beer, Topologies on Closed and Closed Convex Sets. Mathematics and Its Applications 268, Kluwer Academic Publishers, Dordrecht, 1993.

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    $\begingroup$ Thank you very much. I think the book provides it easier. If $X$ is a compact metric space by Corollary 5.1.11, p.143, the Fell topology and the topology induced by the Hausdorff metric coincide on the closed non empty subsets of $X$. In addition by Theorem 5.2.10, p.148, if $X$ is a first countable Hausdorff space (fulfilled if $X$ is a metric space), then the Kuratowski-Painleve limit of sequences coincide with the limit of sequences with respect to the Fell topology. $\endgroup$ – Falrach Aug 9 at 15:06

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