Locally compact spaceIn topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which every point has a compact neighborhood. When locally compact spaces are Hausdorff they are called locally compact Hausdorff, which are of particular interest in mathematical analysis.
Locally normal spaceIn mathematics, particularly topology, a topological spaceX is locally normal if intuitively it looks locally like a normal space. More precisely, a locally normal space satisfies the property that each point of the space belongs to a neighbourhood of the space that is normal under the subspace topology. Formal definitionA topological spaceX is said to be locally normal if and only if each point, x, of X has a neighbourhood that is normal under the subspace topology.
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.Locally Hausdorff spaceIn mathematics, in the field of topology, a topological space is said to be locally Hausdorff if every point has a neighbourhood that is a Hausdorff space under the subspace topology.
Locally connected spaceIn topology and other branches of mathematics, a topological spaceX is locally connected if every point admits a neighbourhood basis consisting of openconnected sets. As a stronger notion, the space X is locally path connected if every point admits a neighbourhood basis consisting of open path connected sets. BackgroundThroughout the history of topology, connectedness and compactness have been two of the most widely studied topological properties.
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.
Maximum and minimumIn mathematical analysis, the maximum and minimum of a function are, respectively, the greatest and least value taken by the function. Known generically as extrema, they may be defined either within a given range (the local or relative extrema) or on the entire domain (the global or absolute extrema) of a function. Pierre de Fermat was one of the first mathematicians to propose a general technique, adequality, for finding the maxima and minima of functions.
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.Neighbourhood systemIn topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter N ( x ) {\displaystyle {\mathcal {N}}(x)} for a point x {\displaystyle x} in a topological space is the collection of all neighbourhoods of x .
Locally normal spaceIn mathematics, particularly topology, a topological spaceX is locally normal if intuitively it looks locally like a normal space. More precisely, a locally normal space satisfies the property that each point of the space belongs to a neighbourhood of the space that is normal under the subspace topology. Formal definitionA topological spaceX is said to be locally normal if and only if each point, x, of X has a neighbourhood that is normal under the subspace topology.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.Locally Hausdorff spaceIn mathematics, in the field of topology, a topological space is said to be locally Hausdorff if every point has a neighbourhood that is a Hausdorff space under the subspace topology.Locally connected spaceIn topology and other branches of mathematics, a topological spaceX is locally connected if every point admits a neighbourhood basis consisting of openconnected sets. As a stronger notion, the space X is locally path connected if every point admits a neighbourhood basis consisting of open path connected sets. BackgroundThroughout the history of topology, connectedness and compactness have been two of the most widely studied topological properties.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.Maximum and minimumIn mathematical analysis, the maximum and minimum of a function are, respectively, the greatest and least value taken by the function. Known generically as extrema, they may be defined either within a given range (the local or relative extrema) or on the entire domain (the global or absolute extrema) of a function. Pierre de Fermat was one of the first mathematicians to propose a general technique, adequality, for finding the maxima and minima of functions.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.Neighbourhood systemIn topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter N ( x ) {\displaystyle {\mathcal {N}}(x)} for a point x {\displaystyle x} in a topological space is the collection of all neighbourhoods of x .*As an Amazon Associate I earn from qualifying purchases.