Ad
related to: topological space ppt presentation
Search results
Results From The WOW.Com Content Network
In mathematics, a topological space is, roughly speaking, a geometrical 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 ...
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest homotopy group.
A three-dimensional model of a figure-eight knot.The figure-eight knot is a prime knot and has an Alexander–Briggs notation of 4 1.. Topology (from the Greek words τόπος, 'place, location', and λόγος, 'study') is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling ...
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space. Intuitively, homotopy groups record information about the basic shape, or holes, of a topological space.
A topological space is called simply connected if it is path-connected and any loop in defined by : can be contracted to a point: there exists a continuous map : such that restricted to is . Here, S 1 {\displaystyle S^{1}} and D 2 {\displaystyle D^{2}} denotes the unit circle and closed unit disk in the Euclidean plane respectively.
The following is a list of named topologies or topological spaces, many of which are counterexamples in topology and related branches of mathematics. This is not a list of properties that a topology or topological space might possess; for that, see List of general topology topics and Topological property.
More briefly, every irreducible closed set has a unique generic point. Any Hausdorff space must be sober, and any sober space must be T 0. X is weak Hausdorff if, for every continuous map f to X from a compact Hausdorff space, the image of f is closed in X. Any Hausdorff space must be weak Hausdorff, and any weak Hausdorff space must be T 1.
The term symmetric space also has another meaning.) A topological space is a T 1 space if and only if it is both an R 0 space and a Kolmogorov (or T 0) space (i.e., a space in which distinct points are topologically distinguishable). A topological space is an R 0 space if and only if its Kolmogorov quotient is a T 1 space.