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A topological space is the most general type of a mathematical space that allows for the definition of limits, continuity, and connectedness. [1][2] Common types of topological spaces include Euclidean spaces, metric spaces and manifolds. Although very general, the concept of topological spaces is fundamental, and used in virtually every branch ...
In the mathematicalfield of topology, a topological spaceis usually defined by declaring its open sets.[1] 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 ...
If a topological space is endowed with the discrete topology (so that by definition, every subset of is open) then every subset of is a clopen subset. For a more advanced example reminiscent of the discrete topology, suppose that U {\displaystyle {\mathcal {U}}} is an ultrafilter on a non-empty set X . {\displaystyle X.}
For every topological space Y, the projection is a closed mapping [11] (see proper map). Every open cover linearly ordered by subset inclusion contains X. [12] Bourbaki defines a compact space (quasi-compact space) as a topological space where each filter has a cluster point (i.e., 8. in the above). [13]
A topological space is a set endowed with a structure, called a topology, which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity. Euclidean spaces, and, more generally, metric spaces are examples of topological spaces, as any distance or metric defines a topology.
Computable topology. Computable topology is a discipline in mathematics that studies the topological and algebraic structure of computation. Computable topology is not to be confused with algorithmic or computational topology, which studies the application of computation to topology.
Computational topology. Algorithmic topology, or computational topology, is a subfield of topology with an overlap with areas of computer science, in particular, computational geometry and computational complexity theory. A primary concern of algorithmic topology, as its name suggests, is to develop efficient algorithms for solving problems ...
Functor. In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and maps between these algebraic objects are associated to continuous maps between spaces. Nowadays ...