Hausdorff spaceIn topology and related branches of mathematics, a Hausdorff space ( , ), T2 space or separated space, is a topological space where distinct points have disjointneighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" (T2) is the most frequently used and discussed. It implies the uniqueness of limits of sequences, nets, and filters. Hausdorff spaces are named after Felix Hausdorff, one of the founders of topology.
T1 spaceIn topology and related branches of mathematics, a T1 space is a topological space in which, for every pair of distinct points, each has a neighborhood not containing the other point. An R0 space is one in which this holds for every pair of topologically distinguishable points. The properties T1 and R0 are examples of separation axioms.
Urysohn's lemmaIn topology, Urysohn's lemma is a lemma that states that a topological space is normal if and only if any two disjointclosed subsets can be separated by a continuous function. Urysohn's lemma is commonly used to construct continuous functions with various properties on normal spaces. It is widely applicable since all metric spaces and all compactHausdorff spaces are normal. The lemma is generalised by (and usually used in the proof of) the Tietze extension theorem.
Separation axiomIn topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider. Some of these restrictions are given by the separation axioms. These are sometimes called Tychonoff separation axioms, after Andrey Tychonoff.
Separated setsType of relation for subsets of a topological space In topology and related branches of mathematics, separated sets are pairs of subsets of a given topological space that are related to each other in a certain way: roughly speaking, neither overlapping nor touching. The notion of when two sets are separated or not is important both to the notion of connected spaces (and their connected components) as well as to the separation axioms for topological spaces.
Closed setThe boundary of the unit disc (pink) is the unit circle (blue). The open disc ‖x‖<1{\displaystyle \|x\|<1} does not include the boundary ‖x‖=1{\displaystyle \|x\|=1}, while the closed unit disc ‖x‖≤1{\displaystyle \|x\|\leq 1} does include the boundary and is therefore a closed set.
Topological spaceIn mathematics, a topological space is, roughly speaking, a space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some axioms formalizing the concept of closeness.
Disjoint setsIn set theory in mathematics and formal logic, two sets are said to be disjoint sets if they have no element in common. Equivalently, two disjoint sets are sets whose intersection is the empty set. For example, {1, 2, 3} and {4, 5, 6} are disjoint sets, while {1, 2, 3} and {3, 4, 5} are not disjoint. A collection of two or more sets is called disjoint if any two distinct sets of the collection are disjoint.
TopologyTopology (from the Greek words , 'place, location', and λόγος, 'study') is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuousdeformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself.
Neighbourhood (mathematics)In topology and mathematical analysis, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. It is closely related to the concepts of open set and interior. Intuitively speaking, a neighbourhood of a point is a set of points containing that point where one can move some amount in any direction away from that point without leaving the set.
MathematicsMathematics is a field of knowledge concerned with abstract concepts such as numbers, geometric shapes, sets, functions, and probabilities. It uses logical reasoning and proof to study and establish their properties, often expressed as theorems, formulas, and equations. Mathematics is used to model and solve problems in science, engineering, technology, economics, and everyday life.*As an Amazon Associate I earn from qualifying purchases.