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 ...
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 ...
The axiomatic approach, which was developed in 1945, allows one to prove results, such as the Mayer–Vietoris sequence, that are common to all homology theories satisfying the axioms. [ 1 ] If one omits the dimension axiom (described below), then the remaining axioms define what is called an extraordinary homology theory .
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.
It's not enough for elements of a topological space to be distinct (that is, unequal); we may want them to be topologically distinguishable. Similarly, it's not enough for subsets of a topological space to be disjoint; we may want them to be separated (in any of various ways). The separation axioms all say, in one way or another, that points or ...
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).
The homotopy hypothesis asks whether a space is something fundamentally algebraic. If one prefers to work with a space instead of a pointed space, there is the notion of a fundamental groupoid (and higher variants): by definition, the fundamental groupoid of a space X is the category where the objects are the points of X and the morphisms are ...