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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 .
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 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 .
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 ...
Let X be a topological space. Then two points x and y in X are topologically distinguishable if they do not have exactly the same neighbourhoods (or equivalently the same open neighbourhoods); that is, at least one of them has a neighbourhood that is not a neighbourhood of the other (or equivalently there is an open set that one point belongs ...
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.
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In topology and related areas of mathematics, 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 ...