Theorems in topology

theorem in geometric topology that every simply connected, closed 3-manifold is homeomorphic to the 3-sphere

board game

theorem about metric spaces

theorem stating that a closed curve divides the plane into two regions

theorem in topology and functional analysis

every continuous function on a compact set has a fixed point

theorem

lemma that a topological space is normal iff any 2 disjoint closed subsets can be separated by a continuous function

theorem

topological space that is homeomorphic to a metric space

theorem that a function f: S→Pow(S) on a compact nonempty convex subset S⊂ℝⁿ, whose graph is closed and whose image f(x) is nonempty and convex for all x∈S, has a fixed point

theorem that closed 3-manifolds uniquely decompose into pieces with 1 of 8 types of geometric structure

theorem that any three objects in space can be simultaneously bisected by a plane

theorem that continuous functions on a closed subset of a normal topological space can be extended to the entire space, preserving boundedness if necessary

given a cover of a metric space, a positive real number δ such that any set of diameter smaller than δ is a subset of a member of the cover

theorem in topology about homeomorphic subsets of Euclidean space

problem in geometric topology

mathematical theorem

three-dimensional smooth curves with small total curvature must be unknotted

Result about foliation of compact 3-manifolds

theorem stating that a complete, simply-connected, n-dimensional Riemannian manifold with sectional curvature taking values in the interval (1, 4] is homeomorphic to the n-sphere