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
The area of Topological Data Analysis (TDA) has emerged recently as being that part of Computational Topology concerned with applying the methods of that subject to the analysis of data sets that are often of very large size; the methods used are adapted from algebraic topology and differential topology and are closely related to those used for spatial reconstruction from scanned data in Visualisation, but the context is, theoretically, not limited to low dimensions nor to data of spatial origin nor, initially, to the visualisation of the data. Its aim, rather, is to give qualitative information on the data, allowing for statistical variation, noise etc.
Gunnar Carlsson, Topology and data, Bull. Amer. Math. Soc. 46 (2009), 255-308 (doi:10.1090/S0273-0979-09-01249-X)
Afra Zomorodian, Topological data analysis, In: Advances in Applied and Computational Topology, Proc. Symp. Applied Math vol 70, 2011 (ams:psapm-70)
Gunnar Carlsson, Persistent Homology and Applied Homotopy Theory, in: Handbook of Homotopy Theory, CRC Press, 2019 (arXiv:2004.00738)
See also
Relation to quantum computing:
Application of topological data analysis (persistent homology) to
analysis of quasicrystals:
analysis of cosmological structure formation:
to analysis of phase transitions:
The suggestion to regard cobordism theory of iso-hypersurfaces, and thus Pontryagin's theorem in Cohomotopy, as a tool in (persistent) topological data analysis (improving on homologuical well groups):
Peter Franek, Marek Krčál, On Computability and Triviality of Well Groups, Discrete Comput Geom (2016) 56: 126 (arXiv:1501.03641, doi:10.1007/s00454-016-9794-2)
Peter Franek, Marek Krčál, Persistence of Zero Sets, Homology, Homotopy and Applications, Volume 19 (2017) Number 2 (arXiv:1507.04310, doi:10.4310/HHA.2017.v19.n2.a16)
Peter Franek, Marek Krčál, Cohomotopy groups capture robust Properties of Zero Sets via Homotopy Theory, talk at ACAT meeting 2015 (pfd slides)
Peter Franek, Marek Krčál, Hubert Wagner, Solving equations and optimization problems with uncertainty, J Appl. and Comput. Topology (2018) 1: 297 (arxiv:1607.06344, doi:10.1007/s41468-017-0009-6)
Last revised on May 14, 2021 at 11:59:31. See the history of this page for a list of all contributions to it.