Ultralimit

In mathematics, an ultralimit is a geometric construction that assigns to a sequence of metric spaces X_n a limiting metric space. The notion of an ultralimit captures the limiting behavior of finite configurations in the spaces X_n and uses an ultrafilter to avoid the process of repeatedly passing to subsequences to ensure convergence. An ultralimit is a generalization of the notion of Gromov–Hausdorff convergence of metric spaces.

Recall that an ultrafilter ω on the set of natural numbers ℕ is a set of subsets of ℕ (whose inclusion function can thought of as a measure) which is closed under finite intersection, upwards-closed, and which, given any subset X of ℕ, contains either X or ℕ∖ X. An ultrafilter ω on ℕ is non-principal if it contains no finite set.

Let ω be a non-principal ultrafilter on ${\displaystyle \mathbb {N} }$. If ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ is a sequence of points in a metric space (X,d) and x∈ X, the point x is called the ω -limit of x_n, denoted ${\displaystyle x=\lim _{\omega }x_{n}}$, if for every ${\displaystyle \epsilon >0}$ we have:

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