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.
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.
If is Sierpinski space (two points , and three opens , , and ), then the continuous map taking to and to is a weak homotopy equivalence1.
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). The isomorphism on the right says that any finite topological space can be constructed by gluing together copies of Sierpinski space, in exactly the same way that any preorder can be constructed by gluing together copies of the preorder .
On the other hand, any finite simplicial complex is homotopy equivalent to its barycentric subdivision, which is the geometric realization of the poset of simplices ordered by inclusion. Thus finite posets model the weak homotopy types of finite simplicial complexes.
A survey is in
The original results by McCord are 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 . ↩
Revision on November 13, 2013 at 09:35:51 by Urs Schreiber See the history of this page for a list of all contributions to it.