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.
The fiber bundle construction theorem is a fundamental result in differential geometry and algebraic topology that provides a way to construct fiber bundles from certain types of spaces. A fiber bundle is a structure that consists of a total space, a base space, a projection map, and a typical fiber that is consistent across the base space. While the theorem itself can be stated in several ways depending on context, it generally concerns the relationship between certain types of spaces and their ability to form fiber bundles under specific conditions.
The Federer-Morse theorem is a result in geometric measure theory that relates to the study of properties of measures in Euclidean space. Specifically, it deals with rectifiable sets and their measures, providing a foundational understanding of how these sets can be characterized and analyzed.
Blumberg's theorem is a result in the field of mathematical analysis, particularly in the area of measure theory. It provides a criterion for a subset of a complete metric space to be measurable. More specifically, the theorem states that in a complete metric space, if a subset is a countable union of closed sets, it is measurable if it is "small" in a certain sense—specifically, if it has a "density" that approaches 1 in certain limits.
The Bing metrization theorem is a result in the field of topology, specifically in the area concerning the metrization of topological spaces. It provides a condition under which a topological space can be given a metric that generates the same topology. Formulated by the mathematician R. Bing in the mid-20th century, the theorem states that if a topological space is second countable and Hausdorff, then it can be metrized.
The Bagpipe Theorem is a concept in the field of mathematical physics, particularly in the study of optimal shapes and configurations. It is often discussed in the context of optimization problems involving geometric shapes and volumes. The theorem essentially deals with the question of how to shape a region or object to maximize or minimize certain properties, such as surface area or volume, while adhering to specific constraints.
The Andreotti–Vesentini theorem is a result in complex geometry concerning the compactness and structure of certain types of complex analytic spaces, particularly in the context of complex manifolds and their cohomological properties. More specifically, it deals with the conditions under which a certain class of complex manifolds (often those with some form of controlled singularities or specific types of curvature) can be compactified or embedded in projective space.
The Anderson–Kadec theorem is a result in the field of functional analysis and specifically in the study of Banach spaces. It addresses the embedding of certain types of Banach spaces into weakly* compact convex sets.
Tarski's undefinability theorem is a result in mathematical logic that deals with the concept of truth within formal languages. Named after the logician Alfred Tarski, the theorem asserts that the notion of truth cannot be defined within a sufficiently expressive formal language that can express arithmetic truths about itself.
The Szpilrajn extension theorem, also known as the Szpilrajn-Sierpiński extension theorem, is a result in order theory, specifically within the area concerning partially ordered sets (posets). The theorem provides a method for extending a given partial order to a total order.
The Schröder–Bernstein theorem is a fundamental result in set theory that provides a criterion for the existence of a bijection (one-to-one and onto correspondence) between two sets, given certain conditions about the existence of injections (one-to-one functions) between those sets. In the context of measurable spaces, the theorem can be reformulated to pertain to the measurability of the functions involved.
Robinson's joint consistency theorem is a result in the field of decision theory and economics related to the consistency of preferences and the representation of preferences by a utility function. The theorem addresses the question of how to represent preferences over a set of choices that may vary according to certain parameters. Specifically, it deals with the conditions under which a joint distribution of choices can be consistent with the preferences of agents when making those choices.