In mathematics, an ultralimit is a geometric construction that assigns a limit metric space to a sequence of metric spaces ${\displaystyle X_{n}}$. The concept captures the limiting behavior of finite configurations in the ${\displaystyle X_{n}}$ spaces employing an ultrafilter to bypass the need for repeated consideration of subsequences to ensure convergence. Ultralimits generalize Gromov–Hausdorff convergence in metric spaces.

An ultrafilter, denoted as ω, on the set of natural numbers ${\displaystyle \mathbb {N} }$ is a set of nonempty subsets of ${\displaystyle \mathbb {N} }$ (whose inclusion function can be thought of as a measure) which is closed under finite intersection, upwards-closed, and also which, given any subset X of ${\displaystyle \mathbb {N} }$, contains either X or ${\displaystyle \mathbb {N} \setminus X.}$ An ultrafilter on ${\displaystyle \mathbb {N} }$ is non-principal if it contains no finite set.

In the following, ω is 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, then the point x is called ω-limit of x_n, denoted as ${\displaystyle x=\lim _{\omega }x_{n}}$, if for every ${\displaystyle \epsilon >0}$ it holds that

It is observed that,

A fundamental fact states that, if (X,d) is compact and ω is a non-principal Ultrafilter on ${\displaystyle \mathbb {N} }$, the ω-limit of any sequence of points in X exists (and is necessarily unique).

In particular, any bounded sequence of real numbers has a well-defined ω-limit in ${\displaystyle \mathbb {R} }$, as closed intervals are compact.

Let ω be a non-principal ultrafilter on ${\displaystyle \mathbb {N} }$. Let (X_n ,d_n) be a sequence of metric spaces with specified base-points p_n ∈ X_n.