Examples/classes:
Types
Related concepts:
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
Given a manifold $\Sigma$, such as the 3-sphere, its space of knots is a topological space parametrizing knots in $\Sigma$, hence something like the topological subspace
of the space of maps $S^1 \to \Sigma$ on those maps which are embeddings. This space is often denoted just $\mathcal{K}$.
A knot invariant is then equivalently a locally constant function on the space of knots.
Allen Hatcher, Spaces of Knots (arXiv:math/9909095)
Ryan Budney, Topology of spaces of knots in dimension 3, Proceedings of the London Mathematical Society Vol 101 pt. 2 (Sept. 2010) (arXiv:math/0506524)
Discussion of the rational homotopy type of knot spaces in terms of graph complexes:
Dev Sinha, The Topology of Spaces of Knots: Cosimplicial Models, American Journal of Mathematics, Vol. 131, No. 4 (Aug., 2009), pp. 945-980 (36 pages) (arXiv:math/0202287, jstor:40263760)
Alberto Cattaneo, Paolo Cotta-Ramusino, Riccardo Longoni, Configuration spaces and Vassiliev classes in any dimension, Algebr. Geom. Topol. 2 (2002) 949-1000 (arXiv:math/9910139)
Alberto Cattaneo, Paolo Cotta-Ramusino, Riccardo Longoni, Algebraic structures on graph cohomology, Journal of Knot Theory and Its Ramifications, Vol. 14, No. 5 (2005) 627-640 (arXiv:math/0307218, doi:10.1142/S0218216505004019, math.GT/0307218, MR2006g:58021)
Pascal Lambrechts, Victor Tourtchine, Homotopy graph-complex for configuration and knot spaces, Transactions of the AMS, Volume 361, Number 1, January 2009, Pages 207–222 (arxiv:math/0611766)
Ismar Volić, Section 4 of: Configuration space integrals and the topology of knot and link spaces, Morfismos, Vol 17, no 2, 2013 (arxiv:1310.7224)
Last revised on October 5, 2019 at 12:22:46. See the history of this page for a list of all contributions to it.