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In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets. [1] [2] [3] That is, a function : is open if for any open set in , the image is open in . Likewise, a closed map is a function that maps closed sets to closed sets.
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
Open Problems in Mathematics. Springer International Publishing. ISBN 978-3-319-32162-2; Guy, R. (2013). Unsolved Problems in Number Theory. Problem Books in Mathematics. Springer New York. ISBN 978-0-387-26677-0
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 ...
An open neighborhood assignment is a function : where () is an open neighborhood. A topological space X {\displaystyle X} is a D-space if for every given neighborhood assignment N x : X → N ( X ) {\displaystyle N_{x}:X\to N(X)} , there exists a closed discrete subset D {\displaystyle D} of the space X {\displaystyle X} such that ⋃ x ∈ D N ...
Informally, a topological property is a property of the space that can be expressed using open sets. A common problem in topology is to decide whether two topological spaces are homeomorphic or not. To prove that two spaces are not homeomorphic, it is sufficient to find a topological property which is not shared by them.
Open problems in continuum theory Examples in continuum theory Continuum Theory and Topological Dynamics , M. Barge and J. Kennedy, in Open Problems in Topology, J. van Mill and G.M. Reed (Editors) Elsevier Science Publishers B.V. (North-Holland), 1990.
The term topology was introduced by Johann Benedict Listing in the 19th century, although it was not until the first decades of the 20th century that the idea of a topological space was developed. This is a list of topology topics. See also: Topology glossary; List of topologies; List of general topology topics; List of geometric topology topics