topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
constructive mathematics, realizability, computability
propositions as types, proofs as programs, computational trinitarianism
Computational topology is a relatively new area of study. It emerged in response to topological problems that arise in computer graphics, robotics and planning. There are interactions with dynamical systems, and computational geometry.
computational homology?
Related areas include
See also:
Textbook accounts:
Herbert Edelsbrunner, John L. Harer, Computational Topology, An Introduction, American Mathematical Society (2010) [ISBN:978-0-8218-4925-5, pdf]
Graham Ellis, An Invitation to Computational Homotopy, Oxford University Press (2019) [ISBN:9780198832973, MAA review]
See also:
Constructive$\;$homology groups and homotopy groups:
Julio Rubio, Francis Sergeraert, Constructive Algebraic Topology, Bulletin des Sciences Mathématiques
126 5 (2002) 389-412 [arXiv:math/0111243, doi:10.1016/S0007-4497(02)01119-3]
Kenzo (computer system for computational topology)
Discussion of (equivariant) homotopies and homotopy groups in computational topology:
Martin Čadek, Marek Krčál, Jiří Matoušek, Lukáš Vokřínek, Uli Wagner, Polynomial-time computation of homotopy groups and Postnikov systems in fixed dimension, SIAM J. Comput., 43(5), 1728–1780 (arXiv:1211.3093, doi:10.1137/120899029)
Lukáš Vokřínek, Computing the abelian heap of unpointed stable homotopy classes of maps, Archivum Mathematicum, Volume 49 (2013), No. 5 · (arXiv:1312.2474, doi:10.5817/AM2013-5-359)
Marek Filakovský, Lukáš Vokřínek, Are two given maps homotopic? An algorithmic viewpoint, Found Comput Math (2019) (arXiv:1312.2337, doi:10.1007/s10208-019-09419-x)
Marek Filakovsky, Peter Franek, Uli Wagner, Stephan Zhechev, Computing simplicial representatives of homotopy group elements, J Appl. and Comput. Topology (2018) 2: 177. (arXiv:1706.00380)
and on the extension-problem:
Discussion of Cohomotopy-sets in computational topology:
Martin Čadek, Marek Krčál, Jiří Matoušek, Francis Sergeraert, Lukáš Vokřínek, Uli Wagner, Computing all maps into a sphere, Journal of the ACM, Volume 61 Issue 3, May 2014 Article No. 1 (arxiv:1105.6257)
Lukáš Vokřínek, Decidability of the extension problem for maps into odd-dimensional spheres, Discrete Comput Geom (2017) 57: 1 (arXiv:1401.3758)
Last revised on August 4, 2023 at 10:02:42. See the history of this page for a list of all contributions to it.