Search results
Results From The WOW.Com Content Network
However, this is not necessary, as there are many equivalent axiomatic foundations, each leading to exactly the same concept. For instance, a topological space determines a class of closed sets, of closure and interior operators, and of convergence of various types of objects. Each of these can instead be taken as the primary class of objects ...
The foundation of this science, for a space of any dimension, was created by Henri Poincaré. His first article on this topic appeared in 1894. [5] In the 1930s, James Waddell Alexander II and Hassler Whitney first expressed the idea that a surface is a topological space that is locally like a Euclidean plane.
The main utility of this notion is in the abundance of situations in mathematics where topological heuristics are very effective, but an honest topological space is lacking; it is sometimes possible to find a topos formalizing the heuristic. An important example of this programmatic idea is the étale topos of a scheme. Another illustration of ...
In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (,) is said to be metrizable if there is a metric : [,) such that the topology induced by d is . Metrization theorems are theorems that give sufficient conditions for a topological space to ...
He refers to topological spaces which satisfy all five axioms as T 1-spaces in contrast to the more general spaces which only satisfy the four listed axioms. Indeed, these spaces correspond exactly to the topological T 1 -spaces via the usual correspondence (see below).
Hence the point space of a locally compact connected Laguerre plane is homeomorphic to the cylinder or it is a -dimensional manifold, cf. [64] A large class of -dimensional examples, called ovoidal Laguerre planes, is given by the plane sections of a cylinder in real 3-space whose base is an oval in .
Foundation of category theory: axioms for categories, functors, and natural transformations. 1945: Norman Steenrod–Samuel Eilenberg: Eilenberg–Steenrod axioms for homology and cohomology. 1945: Jean Leray: Founds sheaf theory. For Leray a sheaf was a map assigning a module or a ring to a closed subspace of a topological space.
A generic point of the topological space X is a point P whose closure is all of X, that is, a point that is dense in X. [1]The terminology arises from the case of the Zariski topology on the set of subvarieties of an algebraic set: the algebraic set is irreducible (that is, it is not the union of two proper algebraic subsets) if and only if the topological space of the subvarieties has a ...