Theorems in topology

In topology, theorems are statements that have been proven to be true based on axioms and previously established theorems within the framework of topology. Topology is a branch of mathematics that studies properties of space that are preserved under continuous transformations.

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.

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 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 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.

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 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.

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 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.

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.

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 \( X \).

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.

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.

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.

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.

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.

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 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.

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 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 \( F: X \to \mathbb{R} \).