by Jens Bossaert (Curiosa Mathematica on Tumblr.com)

by Michael Zarichnyi

by Jens Bossaert (Curiosa Mathematica on Tumblr.com)

by Jens Bossaert (Curiosa Mathematica on Tumblr.com)

by Michael Zarichnyi

The Kakutani fixed-point theorem is a fixed-point theorem for set-valued functions. Let S be a non-empty, compact and convex subset of some Euclidean space. If f is a convex-valued self-map of S with closed graph, then f has a fixed point.

by Michael Zarichnyi

Given a solid ball in 3-dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball. This can be also proved for another bounded domains in 3-dimensional space.

by Michael Zarichnyi

by Michael Zarichnyi

Banach-Schauder theorem (open mapping theorem). If a continuous linear operator between Banach spaces is surjective then it is an open map.

by Michael Zarichnyi

Banach contraction principle. Every contraction in a complete metric space has a unique fixed point.

by Michael Zarichnyi

Hahn-Banach theorem, in its geometric form, states that any two disjoint closed convex sets (at least one of them is assumed to be compact) in a locally conves space can be separated with a hyperplane.

by Michael Zarichnyi

Banach-Mazur distance. This is a distance on the set of [isometry classes of] n-dimensional normed spaces.

by Taras Banakh

by Michael Zarichnyi

by Michael Zarichnyi

by Michael Zarichnyi

by Michael Zarichnyi

by Michael Zarichnyi

A fiber bundle (or, in British English, fibre bundle) is a space that is locally a product space, but globally may have a different topological structure.

by Michael Zarichnyi

by Michael Zarichnyi

Fuzzy sets (aka uncertain sets) are somewhat like sets whose elements have degrees of membership; this is described with the aid of a membership function valued in the real unit interval [0, 1].

by Michael Zarichnyi

by Michael Zarichnyi

by Michael Zarichnyi

A class of objects or methods exhibit recursive behavior when they can be defined by two properties: 1) A simple base case (or cases)-a terminating scenario that does not use recursion to produce an answer. 2) A set of rules that reduce all other cases toward the base case.

by Michael Zarichnyi

Real trees (also called R-trees) are a class of metric spaces generalizing simplicial trees. A metric space is a real tree if it is a geodesic space where every triangle is a tripod.

by Michael Zarichnyi

A solenoid is a compact connected topological space (i.e. a continuum) that may be obtained as the inverse limit of an inverse system of topological groups (e.g., circles) and continuous homomorphisms (e.g., squarings).

by Michael Zarichnyi