The Kolmogorov continuity theorem is a fundamental result in the theory of stochastic processes, particularly in the study of Brownian motion and other continuous-time processes. It provides conditions under which a collection of random variables (typically indexed by time) possesses a continuous version, which means that the sample paths of the process can be modified to be continuous with probability one.

Ignatov's theorem refers to a result in the field of functional analysis, particularly concerning the properties of bounded linear operators on Banach spaces. Specifically, it deals with the existence of certain types of fixed points or invariant elements under the action of a non-expansive operator.

Foster's theorem, often discussed in the context of stochastic processes and in particular for Markov chains and Markov decision processes, provides insights into the long-term behavior of certain types of random processes. One common application of Foster's theorem is in the study of Markov chains with continuous state spaces. In its simplest form, Foster's theorem relates to the existence of a stationary distribution for a Markov chain.

The Clark–Ocone theorem is a fundamental result in the theory of stochastic calculus and financial mathematics, particularly in the context of stochastic processes. This theorem provides a way to express a certain class of random variables (specifically, adapted, or predictable functionals of a process) in terms of an integral with respect to a martingale and a stochastic integral.

The Bussgang theorem is a result in signal processing and statistics, named after Julian J. Bussgang, who introduced it in the context of nonlinear systems. The theorem states that if a Gaussian random process is passed through a nonlinear system, the cross-correlation of the output signal with the input signal can be expressed in terms of the correlation of the input signal alone.

The Bruss–Duerinckx theorem is a result in the field of probability theory and mathematical finance, specifically related to the pricing and replication of contingent claims in incomplete markets. It presents conditions under which a contingent claim can be obtained as the limit of portfolios in a given financial market. The theorem states that if a financial market is incomplete, then under certain conditions, there exists an equivalent martingale measure (a probability measure that allows for the pricing of contingent claims).

The Tietze Extension Theorem is a fundamental result in topology, particularly in the context of normal spaces. It states that if \( X \) is a normal topological space and \( A \) is a closed subset of \( X \), then any continuous function \( f: A \to \mathbb{R} \) can be extended to a continuous function \( F: X \to \mathbb{R} \).

The Sphere Theorem is a result in differential geometry that describes the geometric properties of manifolds with certain curvature conditions. Specifically, it pertains to the behavior of Riemannian manifolds that have non-negative sectional curvature. The Sphere Theorem states that if a Riemannian manifold has non-negative sectional curvature and is simply connected, then it is homeomorphic to a sphere.

The Reeb sphere theorem is a result in differential topology that concerns the topology of the level sets of smooth functions on manifolds, particularly in the context of contact topology. The theorem is named after the mathematician George Reeb.

Quillen's Theorems A and B are important results in the field of algebraic topology, particularly in the study of stable homotopy theory and the homotopy theory of categories. ### Quillen's Theorem A Quillen's Theorem A states that for a simplicial set \( X \), if the simplicial set is Kan, then its associated category of simplicial sets has the homotopy type of a CW-complex.

The Phragmén–Brouwer theorem is a result in complex analysis, specifically within the context of the behavior of holomorphic functions. It generalizes the maximum modulus principle and provides conditions under which a holomorphic function can achieve its maximum on the boundary of a domain.

The Pasting Lemma is a concept from topology, particularly within the study of continuous functions and spaces. It primarily deals with the conditions under which continuous functions defined on overlapping subsets can be "pasted" together to form a new continuous function on a larger space.

Novikov's Compact Leaf Theorem is a result in the field of differential topology, particularly in the study of foliations on smooth manifolds. It addresses the existence of compact leaves in a certain class of foliations, which are decompositions of a manifold into disjoint submanifolds called leaves.

Netto's theorem, also known as the Netto criterion or Netto's criterion, is a result in the field of mathematics, particularly in complex analysis and algebra. The theorem provides a criterion for determining the number of roots of a complex polynomial inside a given contour in the complex plane.

The Mostow–Palais theorem is a notable result in the field of differential topology and algebraic topology. It concerns the concept of the deformation retraction of a manifold and provides insight into the relationship between the topology of a space and its smooth structure.

Lebesgue's Number Lemma is a fundamental result in real analysis, particularly in the context of uniform continuity and compactness. It is often used in the field of topology and forms an important part of the theory of measure and integration. The lemma states the following: Let \( \mathcal{U} \) be an open cover of a compact metric space \( X \).

Janiszewski's theorem is a result in the field of topology, specifically concerning the properties of certain kinds of topological spaces. It deals with the concept of continuity and compactness in the context of mapping spaces.

The Ham Sandwich Theorem is a result in geometry that states that given \( d \) measurable sets in \( d \)-dimensional space, it is possible to simultaneously divide all of them into two equal volumes using a single hyperplane.