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 notion of CW-spectrum is the analogue for sequential spectra in Top of the concept of CW-complex for topological spaces.
Just like CW-complexes are cofibrant objects in the classical model structure on topological spaces, so CW-spectra are cofibrant objects in the stable model structure on topological sequential spectra, see below.
A CW-spectrum $X_\bullet\in SeqSpec(Top)$ is a sequential spectrum in Top such that
all component spaces $X_n$ are CW-complexes,
all structure maps $\Sigma X_n \longrightarrow X_{n+1}$ are inclusions of subcomplexes
e.g. (Adams 74, p. 139) Beware that for instance (Switzer 75, def. 8.1) says just “spectrum” for “CW-spectrum”.
For a CW-spectrum $X$ there is a concept of “cell of a spectrum”:
A cell of a CW-spectrum, def. is a cell of one of the components CW-complexes $X_n$, together with all its suspensions in all the higher component spaces $X_{\gt n}$, subject to the condition that the first cell itself is not itself the suspension of a cell in $X_{n-1}$.
This way every CW-spectrum is the union of all its cells in the sense of def. . (e.g. Switzer 75, 8.4).
A CW-spectrum, def. , is called a finite spectrum (or countable spectrum, etc.) if it has finitely many cells (countably many cells) according to def. .
For $n \in \mathbb{Z}$ (possibly negative) define $\mathbb{S}^n$ to be the sequential prespectrum with component spaces
and with structure maps the canonical isomorphisms.
(Lewis-May-Steinberger 86, def. 4.3)
for $n = 0$ then $\mathbb{S}^0 = \mathbb{S} = \Sigma^\infty S^0$ is standard sequential incarnation of the sphere spectrum;
for $n \geq n$ then $\mathbb{S}^n \simeq \Sigma^\infty S^n$ is the suspension spectrum on the n-sphere;
for general $n$ then $\mathbb{S}^n \simeq F_{-n} S^0$ is also known as the $(-n)$th free spectrum on $S^0$.
A cell spectrum is a topological sequential spectrum $X$ realized as the colimit over a sequence of spectra $\ast = X_0 \to X_1 \to X_2 \to X_3 \to \cdots$ such that there are morphisms
with $X_{n+1}= Cone(j_n)$ (the mapping cone).
(rmk.)
A cell spectrum is a CW-spectrum if each attaching map $\Sigma^\infty S^{q_n}\to X_n$ factors through a $X_k \to X_n$ with $k \lt q$.
(e.g. Lewis-May-Steinberger 86, def. 5.1, def. 5.2, Weiss)
There is an obvious category of CW-spectra given as the full subcategory $CWSpec'$ of $SeqSpec(Top)$ consisting of the CW-spectra, ie. a morphism of CW-spectra $f: X \to Y$ is given by levelwise maps $f_n: X_n \to F_n$ that are compatible with the structure maps. In accordance with Switzer 75, 8.9, we call morphisms in this sense functions.
However, often there is a more useful notion of morphisms between CW-spectra.
Let $E$ be a CW-spectrum. A subspectrum is a CW spectrum $F$ such that each $F_n$ is a subcomplex of $E_n$. A subspectrum is cofinal if for each cell $e_n \in E_n$, there is some $m$ such that $\Sigma^m e_n \in F_{n + m}$.
We can then construct the category $CWSpec$ as the localization of $CWSpec'$ at the cofinal inclusions. Since the class of cofinal inclusions admit a calculus of right fractions, the cateogry $CWSpec$ has the following concrete description - a morphism $X \to Y$ in $CWSpec$ is given by a function $\tilde{X} \to Y$, where $\tilde{X}$ is some cofinal subspectrum of $X$, quotiented by the relation that two such morphisms $f: \tilde{X} \to Y$ and $f': \tilde{X}' \to Y$ are considered equivalent if there is some further cofinal subspectrum $\tilde{X}'' \subseteq \tilde{X} \cap \tilde{X'}$ such that $f|_{\tilde{X}'} = f'|_{\tilde{X}''}$.
A sequential spectrum $X\in SeqSpec(Top)_{stable}$ is cofibrant in the stable model structure on topological sequential spectra in particular if all component spaces are cell complexes and all its structure morphisms $S^1 \wedge X_n \to X_{n+1}$ are relative cell complexes. In particular CW-spectra, def. , are cofibrant in $SeqSpec(Top)_{stable}$.
For the proof see there.
For $X\in SeqSpec(Top)_{stable}$ a CW-spectrum, then its standard cylinder spectrum $X \wedge (I_+)$ is a good cylinder object, in that the inclusion
is a cofibration in $SeqSpec(Top)_{stable}$.
See this prop..
The analog of CW-approximation for topological spaces holds true for topological sequential spectra:
For $X \in SeqSpec(Top)$ a topological sequential spectrum, there exists a CW-spectrum $\hat X$ and a stable weak homotopy equivalence
(e.g. Elmendorf-Kriz-May 95, theorem 1.5)
First let $\hat X_0 \longrightarrow X_0$ be a CW-approximation of the component space in degree 0, via this prop.. Then proceed by induction: suppose that for $n \in \mathbb{N}$ a CW-approximation $\phi_{k \leq n} \colon \hat X_{k \leq n} \to X_{k \leq n}$ has been found such that all the structure maps are respected. Consider then the continuous function
Applying that prop. to this function factors it as
Hence we have obtained the next stage of the CW-approximation. The respect for the structure maps is just this factorization property:
A high-powered way to see this is to use the Quillen equivalence between the stable model structure on topological sequential spectra and the stable Bousfield-Friedlander model structure (see there) on sequential spectra in simplicial sets. This implies that a CW-approximation is given by
where ${\vert - \vert} \dashv Sing$ is degreewise the adjunction between geometric realization and forming singular simplicial complex, and $Q$ denotes any cofibrant replacement in the BF-model structure.
Frank Adams, section III.2 and III.3 of Stable homotopy and generalised homology, 1974
Robert Switzer, Algebraic Topology - Homotopy and Homology, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975.
Anthony Elmendorf, Igor Kriz, Peter May, section 1 of Modern foundations for stable homotopy theory, in Ioan Mackenzie James (ed.), Handbook of Algebraic Topology, 1995 Amsterdam: North-Holland, pp. 213–253, (pdf)
Michael Weiss, around pages 43-46 of Homotopy theory and bordism theory (pdf)
Discussion in the generality of equivariant spectra is in
Last revised on September 5, 2016 at 13:17:19. See the history of this page for a list of all contributions to it.