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On this Wiki page, it claims that (in 4th sentences in related concept section)

For compact metric spaces X, Kuratowski convergence coincides with both convergence in Hausdorff metric and Vietoris topology.

I am wondering can anybody give me a reference where it proves this claim? Thank you!

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If you look in Topics on Analysis by Luigi Ambrosio & Paolo Tilli, the full proof is on pages 73 and 74.

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