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
algebraic topology – application of higher algebra and higher category theory to the study of (stable) homotopy theory
homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
This page is about the article
Analysis Situs
Journal de l’École Polytechnique 2 1 (1895) 1–123
original: gallica:12148/bpt6k4337198/f7
English translation by John Stillwell: pdf, pdf
which initiated (cf. Stillwell p. 6) the modern fields of topology (such as in introducing the notion and terminology of homeomorphisms), of algebraic topology (introducing singular homology) and of homotopy theory (introducing the fundamental group).
See also:
Peter Hilton, p. 282 of: Subjective History of Homology and Homotopy Theory, Mathematics Magazine 61 5 (1988) 282-291 [doi:10.2307/2689545]
Dirk Siersma: Poincaré and Analysis Situs, the beginning of algebraic topology, Nieuw Archief voor Wiskunde serie 5 vol 13 no. 3 (2012) 196- 200 [handle:1874/272371, pdf]
John McCleary: A History of Algebraic Topology, talk at GeoToPhyMa-2013, Rabat (2013) [pdf, pdf]
Wikipedia: Analysis Situs_(paper)
Poincaré took both the term and the basic idea of “analysis situs” from a(n unpublished) proposal that Gottfried Leibniz had made right after inventing differential calculus:
Matthew McMillan: Geometric Quantity in Leibniz’s Analysis Situs, PhD thesis, Oxford (2017) [pdf, pdf]
Vincenzo De Risi: Analysis Situs, the Foundations of Mathematics and a Geometry of Space, The Oxford Handbook of Leibniz, Oxford University Press (2018) 247-258 [hal:03059623, doi:10.1093/oxfordhb/9780199744725.013.22]
Last revised on November 21, 2024 at 07:52:29. See the history of this page for a list of all contributions to it.