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A three-dimensional model of a figure-eight knot.The figure-eight knot is a prime knot and has an Alexander–Briggs notation of 4 1.. Topology (from the Greek words τόπος, 'place, location', and λόγος, 'study') is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling ...
In mathematics, Lehrbuch der Topologie (German for "textbook of topology") is a book by Herbert Seifert and William Threlfall, first published in 1934 and published in an English translation in 1980. It was one of the earliest textbooks on algebraic topology, and was the standard reference on this topic for many years. Albert W. Tucker wrote a ...
In mathematics, pointless topology, also called point-free topology (or pointfree topology) and locale theory, is an approach to topology that avoids mentioning points, and in which the lattices of open sets are the primitive notions. [1] In this approach it becomes possible to construct topologically interesting spaces from purely algebraic ...
Pointless topology (also called point-free or pointfree topology) is an approach to topology that avoids mentioning points. The name 'pointless topology' is due to John von Neumann . [ 9 ] The ideas of pointless topology are closely related to mereotopologies , in which regions (sets) are treated as foundational without explicit reference to ...
The product topology on is equal to the initial topology induced by the canonical projections : as ranges over . [4] Consequently, the initial topology on induced by {:} is equal to the inverse image of the product topology on by the evaluation map:. [4] Furthermore, if the maps {} separate points on then the evaluation map is a homeomorphism ...
Topological geometry deals with incidence structures consisting of a point set and a family of subsets of called lines or circles etc. such that both and carry a topology and all geometric operations like joining points by a line or intersecting lines are continuous.
Topology is the field concerned with the properties of continuous mappings, [101] and can be considered a generalization of Euclidean geometry. [102] In practice, topology often means dealing with large-scale properties of spaces, such as connectedness and compactness .
In topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider. Some of these restrictions are given by the separation axioms. These are sometimes called Tychonoff separation axioms, after Andrey Tychonoff.