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
The Moore space $M(G, n)$, where $G$ is an abelian group and $n \geq 1$, is a topological space which has non-trivial (reduced) homology group $G$ precisely only in dimension $n$ and is simply connected if $n \geq 2$.
(This is somewhat dual to the notion of Eilenberg-MacLane space, which instead has nontrivial homotopy group in one single dimension.)
(the following is based on Hatcher)
Let $G$ be an abelian group. Take a presentation of $G$ i.e. a short exact sequence
where $F$ a free abelian group. Since $H=ker(F\rightarrow G)$, it is a free abelian group as well and we choose bases $\{f_\alpha\}_\alpha$ for $F$ and $\{h_\beta\}_\beta$ for $H$. We then construct a CW complex
by taking the $n$-th skeleton to be $X^n:=\vee_\alpha S^n_\alpha$ and for each $\beta$ we attach an $n+1$-cell as follows:
Write $h_\beta=\Sigma_\alpha d_{\alpha\beta}f_\alpha$ and let $\delta_{d_{\alpha\beta}}$ be $0$ if $d_{\alpha\beta}= 0$ and be $1$ otherwise. Define an attaching map $S^n_\beta\rightarrow X^n$ by contracting $\ell_\beta:=(\Sigma_\alpha \delta_{d_{\alpha\beta}})-1$ $(n-1)-$spheres in $S^n$ thus defining a map $S^n_\beta\rightarrow \vee_{\ell_\beta} S^n_{\alpha\beta}$ and then map each $S^n_{\alpha\beta}$ to $S^n_{\alpha}$ by a degree $d_{\alpha\beta}$.
The (homotopy type of) the topological space $M(G,n)$ constructed this way we call the Moore space of $G$ in degree $n$.
The resulting CW-complex can be seen to have the desired properties via cellular homology.
The homotopy type of $M(G,n)$ is determined by specifying $G$ and $n$.
The construction above is not functorial in $G$ because of the choice of bases (see more below). However, it does give a functor to the homotopy category $M(-,n):Ab\rightarrow Ho(Top)$.
The functoriality problem of the construction above cannot be corrected. That is, there is no functor $Ab\rightarrow Top$ that lifts $M(-,n)$. This can be seen as a corollary of a counterexample of Carlsson which gives a negative answer to a conjecture of Steenrod:
(Steenrod)
Given a group $G$ a $G$-module $M$ and a natural number $n$, there is a $G$-space $X$ which has only one non-zero reduced homology G-module in dimension $n$ that satisfy $\tilde{H}_n(X;\mathbb{Z}) \cong M$ as $G$-modules.
Carlsson provides counter examples for such “equivariant Moore spaces” for all non-cyclic groups.
There is thus no functor Ab$\rightarrow$ Top that lifts $M(-,n)\colon Ab\rightarrow Ho(Top)$ since if there was such, it would induce, for any group $G$ a functor $Ab^G\rightarrow Top^G$ and in particular a positive answer to the Steenrod conjecture.
Moreover, there can also not be an (∞,1)-functor $Ab\rightarrow L_{whe} Top$ that lifts $M(-,n)$ since this will similarly yield an $\infty$-functor $Ab^G\rightarrow Top^{hG}$ where $Top^{hG}$ is the (∞,1)-category of ∞-actions of $G$ on spaces. Since there is a “rigidification” functor $Top^{hG}\rightarrow Top^G$ this would yield an (ordinary) functor $Ab^G\rightarrow Top^G$ which does not exist by our previous observation.
There is also a cohomology analogue known as a co-Moore space or a Peterson space, but this is not defined for all abelian $G$. Spheres are both Moore and co-Moore spaces for $G = \mathbb{Z}$.
Co-Moore spaces are the Eckmann–Hilton duals of Eilenberg–Mac Lane spaces.
According to Baues, Moore spaces are $H \pi$-duals to Eilenberg–Mac Lane spaces. This leads to an extensive duality for connected CW complexes.
Just as there is a Postnikov decomposition of a space as a tower of fibrations, so there is a Moore decomposition as a tower of cofibrations.
Marek Golasinski and Daciberg Lima Gonçalves?, On Co-Moore Spaces
Hans J. Baues, Homotopy types, in Handbook of Algebraic Topology, (edited by I.M. James), North Holland, 1995.
Gunnar Carlsson “A counterexample to a conjecture of Steenrod” Invent. Math. 64 (1981), no. 1, 171–174.
Last revised on April 5, 2021 at 02:07:59. See the history of this page for a list of all contributions to it.