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 filtered topological space is a filtered object in Top, hence
A filtered space is called a connected filtered space if it satisfies the following condition:
: The function induced by inclusion is surjective for all ; and, for all , for all and .
There are two other forms of this condition which are useful under different circumstances.
A CW-complex with its filtration by skeleta .
A configuration space of points is filtered by number of points.
The free topological monoid on a topological space filtered by the length of words. Given a pointed topological space , there is also a reduced version by taking and identifying with the identity of . This latter filtered space is known as the James construction (James 55).
The James construction may be constructed homotopy-theoretically (Brunerie 13, Brunerie 17). Recall that, for a finite simplicial complex, for a pair of spaces, its polyhedral product is defined as the union as a subspace of the Cartesian product . Here, for a simplex of , the subspace consist of those such that, for each vertex in the complement of , the coordinate projection lies in . Equivalently, the polyhedral product can be considered as a homotopy colimit of these over the poset of simplexes of , where the maps are the respective inclusions.
For a space equipped with a basepoint , define a filtered space as follows. Set as . For , require that the following square is homotopy pushout:
where the unlabeled arrow is the homotopy colimit of a morphism of diagrams over given by the maps
for each simplex of the boundary simplicial complex of the standard -simplex.
For a pointed space, if is path-connected, then .
A similar example to the last using free groups instead of free monoids.
A similar example to the last using free groupoids on topological graphs.
A similar example to the last using the universal topological groupoid induced from a topological groupoid by a continuous function to a space .
Examples of connected filtered spaces are:
The skeletal filtration of a CW-complex.
The word length filtration of the James construction for a space with base point such that is a closed cofibration.
The filtration of the classifying space of a crossed complex, filtered using skeleta of .
This condition occurs in the higher homotopy van Kampen theorem for crossed complexes.
We thus see that filtered spaces arise from many geometric and algebraic situations, and see also stratified spaces). It is therefore interesting that one can define strict higher homotopy groupoids for filtered spaces more easily than for spaces themselves.
Note also that it is standard to be able to replace, using mapping cylinders, a sequence of maps by a sequence of inclusions.
The James construction is due to
Review of the James construction includes
Dylan Wilson, James construction, 2017 (pdf)
Wikipedia, James reduced product
Discussion of the James construction via homotopy type theory includes the following
Guillaume Brunerie, The James Construction and , talk at the Institute of Advanced Studies on March 27, 2013 (recording)
Guillaume Brunerie, The James construction and in homotopy type theory (arXiv:1710.10307)
Last revised on May 7, 2023 at 14:53:59. See the history of this page for a list of all contributions to it.