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
In cosmology, by cosmic topology one refers to (the possibility, or possible detection of) non-trivial global topology of the spacetime manifold which is the cosmos, hence its (possible) non-homeomorphy to the Cartesian space with its Euclidean topology which is underlying Minkowski spacetime.
Cosmic topology is usually understood to refer to the universe at large only: for instance any non-trivial topology of black hole spacetimes (due to being defined only on the complement spacetime of their would-be singularity, cf. singular brane) is not of concern in discussions of cosmic topology.
For example, in a simple scenario of cosmic topology, space at large might be homeomorphic to a 3-torus of large (huge) radii (which could in principle be detected by showing that images of galaxies repeat periodically in certain directions) or to a 3-sphere (cf. here), or to a lens space [URLW 2004, ALS 2005, Aurich & Lustig 2012a, 2012b], which might be detectable by cose analysis of the cosmic microwave background.
Of particular interest for cosmic electromagnetism would be non-simply-connected cosmic topologies (such as the 3-torus or a lens space but not the 3-sphere) since these could support locally trivial but globally non-trivial electromagnetic fields (namely with vanishing electromagnetic flux density but still non-trivial holonomy/monodromy around non-conctracticle cosmological curves) such as discussed in the Aharonov-Bohm effect.
Moreover, lens space-topology with its integral torsion-cohomology group in degree 2 (cf. there) would even support “fractional” such electromagnetic fields (for which certain integer multiples of the field would vanish), which would in principle be detectable by an uncertainty principle for quantum observables on the cosmic electromagnetic field [Freed, Moore & Segal 2007b, p. 28].
Marc Lachièze-Rey, Jean-Pierre Luminet, Cosmic Topology, Phys. Rept. 254 (1995) 135-214 [arXiv:gr-qc/9605010, doi:10.1016/0370-1573(94)00085-H]
Jean-Pierre Luminet, Boudewijn F. Roukema, Topology of the Universe: Theory and Observations, in: Theoretical and Observational Cosmology, NATO Science Series 541, Springer (1999) 117-152 [doi:10.1007/978-94-011-4455-1_2]
G. T. Gomero, M. J. Reboucas, R. Tavakol, Detectability of Cosmic Topology in Almost Flat Universes, Class. Quant. Grav. 18 (2001) 4461-4476 [arXiv:gr-qc/0105002, doi:10.1088/0264-9381/18/21/306]
Janna Levin, Topology and the Cosmic Microwave Background, Phys. Rept. 365 (2002) 251-333 [arXiv:gr-qc/0108043, doi:10.1016/S0370-1573(02)00018-2]
Neil J. Cornish, David N. Spergel, Glenn D. Starkman, Eiichiro Komatsu, Constraining the Topology of the Universe, Phys. Rev. Lett. 92 (2004) 201302 [arXiv:astro-ph/0310233, doi:10.1103/PhysRevLett.92.201302]
M. J. Rebouças, G. I. Gomero: Cosmic topology: a brief overview, Braz. J. Phys. 34 (4a) (2004) [doi:10.1590/S0103-97332004000700012, pdf]
Boudewijn F. Roukema, Bartosz Lew, Magdalena Cechowska, Andrzej Marecki, Stanislaw Bajtlik, A Hint of Poincaré Dodecahedral Topology in the WMAP First Year Sky Map, Astron. Astrophys. 423 (2004) 821-831 [arXiv:astro-ph/0402608, doi:10.1051/0004-6361:20040337]
Jean-Philippe Uzan, Alain Riazuelo, Roland Lehoucq, Jeffrey Weeks, Cosmic microwave background constraints on lens spaces, Phys. Rev. D 69 (2004) 043003 [doi:10.1103/PhysRevD.69.043003]
Ralf Aurich, Sven Lustig, Frank Steiner, CMB Anisotropy of Spherical Spaces, Class. Quant. Grav. 22 (2005) 3443-3460 [arXiv:astro-ph/0504656, doi:10.1088/0264-9381/22/17/006]
Ralf Aurich, Sven Lustig, Frank Steiner, The circles-in-the-sky signature for three spherical universes, Monthly Notices of the Royal Astronomical Society 369 1 (2006) 240–248 [doi:10.1111/j.1365-2966.2006.10296.x, arXiv:astro-ph/0510847]
Jean-Pierre Luminet, The Shape and Topology of the Universe [arXiv:0802.2236, inspire:779504]
G. F. E. Senden, The Topology of the Universe (2010) [fse:9361, pdf]
Ralf Aurich, Sven Lustig, How well-proportioned are lens and prism spaces?, Class. Quantum Grav. 29 (2012) 175003 [arXiv:1201.6490, doi:10.1088/0264-9381/29/17/175003]
Ralf Aurich, Sven Lustig, A survey of lens spaces and large-scale CMB anisotropy, Mon. Not. Roy. Astron. Soc. 424 (2012) 1556-1562 [arXiv:1203.4086, doi:10.1111/j.1365-2966.2012.21363.x]
Jean-Pierre Luminet: Cosmic Topology, Scholarpedia (2015) [doi:10.4249/scholarpedia.31544]
Jean-Pierre Luminet, The Status of Cosmic Topology after Planck Data, Universe 2 1 (2016) 1 [doi:10.3390/universe2010001, arXiv:1601.03884]
(discussion relative to Planck Data)
Jaspreet Sandhu, Cosmic Topology [arXiv:1612.04157]
Yashar Akrami et al., Promise of Future Searches for Cosmic Topology, Phys. Rev. Lett. 132 17 (2024) 171501 [arXiv:2210.11426, doi:10.1103/PhysRevLett.132.171501]
The COMPACT collaboration:
COMPACT Collaboration: Cosmic topology. Part I. Limits on orientable Euclidean manifolds from circle searches [arXiv:2211.02603]
COMPACT Collaboration: Cosmic topology. Part Ic. Limits on lens spaces from circle searches [arXiv:2409.02226]
COMPACT Collaboration: Cosmic topology. Part IIa. Eigenmodes, correlation matrices, and detectability of orientable Euclidean manifolds [arXiv:2306.17112]
COMPACT Collaboration: Cosmic topology. Part IIIa. Microwave background parity violation without parity-violating microphysics [arXiv:2407.09400]
See also:
Wikipedia, Global universe structure
The Universe’s Topology May Not Be Simple, Physics 17 74 (2024)
Analysis in the generality of inhomogeneous cosmology:
Last revised on November 14, 2024 at 03:17:46. See the history of this page for a list of all contributions to it.