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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 closed unit ball of the dual of a normed vector space over the reals has an extreme point. Point-set topology. The Cartesian product of any family of connected topological spaces is connected. Tychonoff's theorem: The Cartesian product of any family of compact topological spaces is compact.
According to the mathematical physicist John Baez from the University of California, Riverside, The Large Scale Structure of Space–Time was "the first book to provide a detailed description of the revolutionary topological methods introduced by Penrose and Hawking in the early seventies." [4]
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 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 ...
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 set with a topology is called a topological space. Metric spaces are an important class of topological spaces where a real, non-negative distance, also called a metric, can be defined on pairs of points in the set. Having a metric simplifies many proofs, and many of the most common topological spaces are metric spaces.