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A topological space is a set together with a collection of subsets of satisfying: [3]. The empty set and are in .; The union of any collection of sets in is also in .; The intersection of any pair of sets in is also in . Equivalently, the intersection of any finite collection of sets in is also in .
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 .
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 ...
A fiber bundle B is a named set (E, p, B) where the topological space E is the space of B; the topological space B the base of B; and p is a topological projection of E onto B such that every point in B has a neighborhood U such that p −1 (b) = F for all points b from B and p −1 (U) is homeomorphic to the direct product U × F where F is ...
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).
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.