topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
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fiber space, space attachment
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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
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homotopy theory, (∞,1)-category theory, homotopy type theory
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see also algebraic topology
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The free topological monoid $F X$ on a topological space $X$ is canonically filtered by the length of words. Given instead a pointed topological space $(X,x)$, there is also a reduced version by taking $F X$ and identifying $x$ with the identity of $F X$. This latter filtered topological space is known as the James construction $J(X,x)$ (James 55).
The James construction $J(X,x)$ may be constructed homotopy-theoretically (Brunerie 13, Brunerie 17). Recall that, for $K$ a finite simplicial complex, for $(X,A)$ a pair of spaces, its polyhedral product $(X,A)^K$ is defined as the union $\bigcup_{\sigma\in S(K)}(X,A)^\sigma$ as a subspace of the Cartesian product $X^{V(K)}$. Here, for $\sigma\in S(K)$ a simplex of $K$, the subspace $(X,A)^\sigma$ consist of those $x\in X^{V(K)}$ such that, for each vertex $v$ in the complement of $\sigma$, the coordinate projection $\proj_v x$ lies in $A$. Equivalently, the polyhedral product $(X,A)^K$ can be considered as a homotopy colimit of these $(X,A)^\sigma$ over the poset $S(K)$ of simplexes $\sigma$ of $K$, where the maps are the respective inclusions.
For $X$ a space equipped with a basepoint $x$, define a filtered space $fil_\bullet$ as follows. Set $\fil_0$ as $\{x\}$. For $k\ge 1$, require that the following square is homotopy pushout:
where the unlabeled arrow is the homotopy colimit of a morphism of diagrams over $S(\partial \Delta[k-1])$ given by the maps
for each simplex $\sigma$ of the boundary simplicial complex $\partial \Delta[k-1]$ of the standard $(k-1)$-simplex. The homotopy colimit $fil_\infty$ of the sequence of maps $fil_0 \stackrel{j_1}{\to} fil_1 \stackrel{j_2}{\to} \ldots$ is called the James construction on $(X, x)$.
For $(X,x)$ a pointed space, if $(X,x)$ is path-connected, then $fil_\infty \simeq \Omega\Sigma X$.
The James construction of $X$ is homotopy equivalent to the configuration space $C(\mathbb{R}^1, X)$ of points in the real line with “charges” taking values in $X$.
(e.g. Bödigheimer 87, Example 9)
The construction is due to
Review:
Michael Hopkins (notes by Akhil Mathew), Lectures 2,3 in: Spectra and stable homotopy theory, 2012 (pdf, pdf)
Dylan Wilson, James construction, 2017 (pdf)
Wikipedia, James reduced product
Discussion via configuration spaces includes
Discussion via homotopy type theory includes the following
Guillaume BrunerieThe James Construction and $\pi_4(S^3)$, talk at the Institute of Advanced Studies on March 27, 2013 (recording)
Guillaume Brunerie, The James construction and $\pi_4(S^3)$ in homotopy type theory (arXiv:1710.10307)
Last revised on November 29, 2020 at 15:22:04. See the history of this page for a list of all contributions to it.