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
A finite topological space is a topological space whose underlying set is a finite set.
Every finite topological space is an Alexandroff space, i.e. finite topological spaces are equivalent to finite preordered sets, by the specialisation order.
Finite topological spaces have the same weak homotopy types as finite simplicial complexes / finite CW-complexes.
This is due to (McCord 67).
If is Sierpinski space (two points , and three opens , , and ), then the continuous map taking to and to is a weak homotopy equivalence1.
The essential construction in the proof is as follows: for any finite topological space with specialization order , the topological interval map induces a weak homotopy equivalence :
(where we implicitly identify with the category of finite intervals with distinct top and bottom, so that is a covariant functor on ). A few remarks on this construction:
The interval has elements, two of which are the distinct top and bottom. The space is the -dimensional affine simplex. The space has points , where is in the closure of for . The map induced by takes every interior point of to .
Informally, the isomorphism on the right says that any finite topological space can be constructed by gluing together copies of Sierpinski space , just as any preorder can be constructed by gluing together copies of the preorder . More formally, the isomorphism is established for objects in the equivalent category , by restricting an isomorphism over objects of the larger category , given by the counit of a nerve and realization adjunction
where the counit is an isomorphism because the inclusions are fully faithful.
On the other hand, any finite simplicial complex is homotopy equivalent to its barycentric subdivision. This is , the geometric realization of the nerve of the poset whose elements are simplices ordered by inclusion. Thus finite posets model the weak homotopy types of finite simplicial complexes.
A survey which includes the McCord theorems as background material is in
published as
The original results by McCord are in
Michael C. McCord, Singular homology groups and homotopy groups of finite topological spaces , Duke Math. J. 33 (1966), 465-474. (EUCLID)
Michael C. McCord, Homotopy type comparison of a space with complexes associated with its open covers . Proc. Amer. Math. Soc. 18 (1967), 705-708, doi:10.1090/S0002-9939-1967-0216499-0, copy
Generalization to ringed finite spaces is discussed in
Fernando Sancho de Salas, Ringed Finite Spaces (arXiv:1409.4574)
Fernando Sancho de Salas, Finite Spaces and Schemes (arXiv:1602.02393)
and aspects of their homotopy theory is discussed in
Any topological meet-semilattice with a bottom element , for which there exists a continuous path connecting to the top element , is in fact contractible. The contracting homotopy is given by the composite . ↩
Last revised on July 27, 2022 at 22:54:48. See the history of this page for a list of all contributions to it.