Definitions
Let (X, d) be a metric space. For any point x ∈ X and any non-empty compact subset A ⊆ X, let
- .
For any sequence of such subsets An ⊆ X, n ∈ N, the Kuratowski limit inferior (or lower closed limit) of An as n → ∞ is
the Kuratowski limit superior (or upper closed limit) of An as n → ∞ is
If the Kuratowski limits inferior and superior agree (i.e. are the same subset of X), then their common value is called the Kuratowski limit of the sets An as n → ∞ and denoted Ltn→∞An.
The definitions for a general net of compact subsets of X go through mutatis mutandis.
Read more about this topic: Kuratowski Convergence
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