homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
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
For every topological space $X$ there is a CW complex $Z$ and a weak homotopy equivalence $f \colon Z\to X$. Such a map $f \colon Z\to X$ is called a CW approximation to $X$.
Such CW-approximation may be constructed case-by-case by iteratively attaching (starting from the empty space) cells for each representative of a homotopy group of $X$ and further cells to kill off spurious homotopy groups introduced this way (e.g. Hatcher, p. 352-353).
In the classical model structure on topological spaces $Top_{Quillen}$, the cofibrant objects are the retracts of cell complexes, and hence CW approximations are in particular cofibrant replacements in this model structure.
The Quillen equivalence $Top_{Quillen} \stackrel{\overset{{\vert - \vert}}{\longleftarrow}}{\underset{Sing}{\longrightarrow}} sSet_{Quillen}$ to the classical model structure on simplicial sets (“homotopy hypothesis”) yields a functorial CW approximation (by this proposition) via
(geometric realization of the singular simplicial complex of $X$) with the adjunction counit
a weak homotopy equivalence.
Let $f \;\colon\; A \longrightarrow X$ be a continuous function between topological spaces. Then there exists for each $n \in \mathbb{N}$ a relative CW-complex $\hat f \colon A \hookrightarrow \hat Y$ together with an extension $\phi \colon Y \to X$, i.e.
such that $\phi$ is n-connected.
Moreover:
if $f$ itself is k-connected, then the relative CW-complex $\hat f$ may be chosen to have cells only of dimension $k + 1 \leq dim \leq n$.
if $A$ is already a CW-complex, then $\hat f \colon A \to X$ may be chosen to be a subcomplex inclusion.
For every continuous function $f \colon A \longrightarrow X$ out of a CW-complex $A$, there exists a relative CW-complex $\hat f \colon A \longrightarrow \hat X$ that factors $f$ followed by a weak homotopy equivalence
Apply lemma 1 iteratively for $n \in \mathbb{N}$ to produce a sequence with cocone of the form
where each $f_n$ is a relative CW-complex adding cells exactly of dimension $n$, and where $\phi_n$ in n-connected.
Let then $\hat X$ be the colimit over the sequence (its transfinite composition) and $\hat f \colon A \to X$ the induced component map. By definition of relative CW-complexes, this $\hat f$ is itself a relative CW-complex.
By the universal property of the colimit this factors $f$ as
Finally to see that $\phi$ is a weak homotopy equivalence: since n-spheres are compact topological spaces, then every map $\alpha \colon S^n \to \hat X$ factors through a finite stage $i \in \mathbb{N}$ as $S^n \to X_i \to \hat X$ (by this lemma). By possibly including further into higher stages, we may choose $i \gt n$. But then the above says that further mapping along $\hat X \to X$ is the same as mapping along $\phi_i$, which is $(i \gt n)$-connected and hence an isomorphism on the homotopy class of $\alpha$.
For $X$ any sequential spectrum in Top, then there exists a CW-spectrum $\hat X$ and a homomorphism $\phi \colon \hat X \to X$ which is degreewise a weak homotopy equivalence, hence in particular a stable weak homotopy equivalence.
First let $\hat X_0 \longrightarrow X_0$ be a CW-approximation of the component space in degree 0, via prop. 2. 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 prop. 2 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:
Tammo tom Dieck, section 8.6 of Algebraic topology, EMS (2008)
Allen Hatcher, Algebraic topology, Cambridge Univ. Press 2002; Chapter 4, Section 4.1 pdf