topology (point-set topology, point-free topology)
see also 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 CW-complex is a nice topological space which is, or can be, built up inductively, by a process of attaching n-disks $D^n$ along their boundary (n-1)-spheres $S^{n-1}$ for all $n \in \mathbb{N}$: a cell complex built from the basic topological cells $S^{n-1} \hookrightarrow D^n$.
Being, therefore, essentially combinatorial objects, CW complexes are the principal objects of interest in algebraic topology; in fact, most spaces of interest to algebraic topologists are homotopy equivalent to CW-complexes. Notably the geometric realization of every simplicial set, hence also of every groupoid, 2-groupoid, etc., is a CW complex.
Milnor has argued that the category of spaces which are homotopy equivalent to CW-complexes, also called m-cofibrant spaces, is a convenient category of spaces for algebraic topology.
Also, CW complexes are the cofibrant objects in the classical model structure on topological spaces. This means in particular that every topological space is weakly homotopy equivalent to a CW-complex (but need not be strongly homotopy equivalent to one). See also at CW-approximation. Since every topological space is a fibrant object in this model category structure, this means that the full subcategory of Top on the CW-complexes is a category of “homotopically very good representatives” of homotopy types. See at homotopy theory and homotopy hypothesis for more on this.
(origin of the “CW” terminology)
The terminology “CW-complex” goes back to John Henry Constantine Whitehead (see Hatcher, “Topology of cell complexes”, p. 520). It stands for the following two properties shared by any CW complex:
C = “closure finiteness”: a compact subset of a CW complex intersects the interior of only finitely many cells (prop.), hence in particular so does the closure of any cell.
W = “weak topology”: Since a CW-complex is a colimit in Top over its cells, and as such equipped with the final topology of the cell inclusion maps, a subset of a CW-complex is open or closed precisely if its restriction to (the closure of) each cell is open or closed, respectively.
(Whitehead called the interior of the n-disks the “cells”, so that their closure of each cell is the corresponding $n$-disk.)
In the following let Top be the category of topological spaces, or any of its variants, convenient category of topological spaces.
(spheres and disks)
For $n \in \mathbb{N}$ write
$D^n \in Top$ for the n-disk, for instance realized (up to homeomorphism) as the closed unit ball in the $n$-dimensional Euclidean space $\mathbb{R}^n$ and equipped with the induced subspace topology;
$S^{n-1} \in Top$ for the (n-1)-sphere, for instance realized as the boundary of the n-disk, also equipped with the corresponding subspace topology;
$i_n \;\colon\; S^{n-1} \hookrightarrow D^n$ for the continuous function that exhibits this boundary inclusion.
We also call these functions the generating cofibrations (of the classical model structure on topological spaces).
Notice that
$S^{-1} = \emptyset$;
$S^0 = \ast \sqcup \ast$.
(single cell attachment)
For $X$ any topological space and for $n \in \mathbb{N}$, then an $n$-cell attachment to $X$ is the result of gluing an n-disk to $X$, along a prescribed image of its bounding (n-1)-sphere (def. 1):
Let
be a continuous function, then the attaching space
is the topological space which is the pushout of the boundary inclusion of the $n$-sphere along $\phi$, hence the universal space that makes the following diagram commute
If we take the defining boundary inclusion $\iota_n \colon S^{n-1} \to D^n$ itself as an attaching map, then we are gluing two $n$-disks to each other along their common boundary $S^{n-1}$. The result is the $(n+1)$-sphere:
(graphics from Ueno-Shiga-Morita 95)
A single cell attachment of a 0-cell, according to def. 2 is the same as forming the disjoint union space $X \sqcup \ast$ with the point space $\ast$:
In particular if we start with the empty topological space $X = \emptyset$ itself, then by attaching 0-cells we obtain a discrete topological space. To this then we may attach higher dimensional cells.
(attaching many cells at once)
If we have a set of attaching maps $\{S^{n_i-1} \overset{\phi_i}{\longrightarrow} X\}_{i \in I}$ (as in def. 2), all to the same space $X$, we may think of these as one single continuous function out of the disjoint union space of their domain spheres
Then the result of attaching all the corresponding $n$-cells to $X$ is the pushout of the corresponding disjoint union of boundary inclusions:
Apart from attaching a set of cells all at once to a fixed base space, we may “attach cells to cells” in that after forming a given cell attachment, then we further attach cells to the resulting attaching space, and ever so on:
Let $X$ be a topological space, then A topological relative cell complex of countable height based on $X$ is a continuous function
and a sequential diagram of topological space of the form
such that
each $X_k \hookrightarrow X_{k+1}$ is exhibited as a cell attachment according to def. 3, hence presented by a pushout diagram of the form
$Y = \underset{k\in \mathbb{N}}{\cup} X_k$ is the union of all these cell attachments, and $f \colon X \to Y$ is the canonical inclusion; or stated more abstractly: the map $f \colon X \to Y$ is the inclusion of the first component of the diagram into its colimiting cocone $\underset{\longrightarrow}{\lim}_k X_k$:
If here $X = \emptyset$ is the empty space then the result is a map $\emptyset \hookrightarrow Y$, which is equivalently just a space $Y$ built form “attaching cells to nothing”. This is then called just a topological cell complex of countable hight.
Finally, a topological (relative) cell complex of countable hight is called a CW-complex is the $(k+1)$-st cell attachment $X_k \to X_{k+1}$ is entirely by $(k+1)$-cells, hence exhibited specifically by a pushout of the following form:
A finite CW-complex is one which admits a presentation in which there are only finitely many attaching maps, and similarly a countable CW-complex is one which admits a presentation with countably many attaching maps.
Given a CW-complex, then $X_n$ is also called its $n$-skeleton.
A cellular map between CW-complexes $X$ and $Y$ is a continuous function $f\colon X \to Y$ such that $f(X_n) \subset Y_n$.
If $A \hookrightarrow X$ is an inclusion of CW-complexes, then the quotient $X/A$ is naturally itself a CW-complex, such that the quotient map $X \to X/A$ is cellular.
If $X$ is a CW-complex and $K$ is a finite CW-complex, then the product topological space $X \times K$ is naturally itself a CW-complex.
For example the suspension of a CW-complex itself carries the structure of a CW-complex.
Similarly for pointed CW-complexes: the smash product of a pointed CW-complex with a finite pointed CW-complex is a pointed CW-complex. For example the reduced suspension $S^1 \wedge X$ of a pointed CW-complex $X$ is itself a CW-complex.
For $X$ and $Y$ CW-complexes with attaching maps $\{\phi_\alpha\}$ and $\{\Psi_\beta\}$, then the k-ification $(X \times Y)_c$ of their product topological space $X \times Y$ (hence their Cartesian product in the category of compactly generated topological spaces) is again a CW-complex with attaching maps $\{\Phi_\alpha \times \Psi_\beta\}$.
If either of the two CW-complexes is a locally compact topological space or if both are countable CW-complexes (have a countable set of cells) then
and so then the product topological space $X \times Y$ itself has CW-complex structure.
A CW-complex is a locally contractible topological space.
For instance (Hatcher, prop. A.4).
(CW-complexes are paracompact Hausdorff spaces)
Every CW-complex is a
a normal topological space, in particular a Hausdorff space,
Every CW-complex is a compactly generated topological space.
Since a CW-complex $X$ is a colimit in Top over attachments of standard n-disks $D^{n_i}$ (its cells), by the characterization of colimits in $Top$ (prop.) a subset of $X$ is open or closed precisely if its restriction to each cell is open or closed, respectively. Since the $n$-disks are compact, this implies one direction: if a subset $A$ of $X$ intersected with all compact subsets is closed, then $A$ is closed.
For the converse direction, since a CW-complex is a Hausdorff space and since compact subspaces of Hausdorff spaces are closed, the intersection of a closed subset with a compact subset is closed.
Every CW complex is homotopy equivalent to (the realization of) a simplicial complex.
See Gray, Corollary 16.44 (p. 149) and Corollary 21.15 (p. 206). For more see at CW approximation.
Every CW complex is homotopy equivalent to a space that admits a good open cover.
If $Y$ has the homotopy type of a CW complex and $X$ is a finite CW complex, then the mapping space $Y^X$ with the compact-open topology has the homotopy type of a CW complex.
For $X$ a CW complex, the inclusion $X' \hookrightarrow X$ of any subcomplex has an open neighbourhood in $X$ which is a deformation retract of $X'$. In particular such an inclusion is a good pair in the sense of relative homology.
For instance (Hatcher, prop. A.5).
For $A \hookrightarrow X$ the inclusion of a subcomplex into a CW complex, then the pair $(X,A)$ is often called a CW-pair. This appears notably in the axioms for generalized (Eilenberg-Steenrod) cohomology.
e.g. (AGP 02, def. 5.1.11)
We discuss aspects of the singular homology $H_n(-) \colon$ Top $\to$ Ab of CW-complexes. See also at cellular homology of CW-complexes.
Let $X$ be a CW-complex and write
for its filtered topological space-structure with $X_{n+1}$ the topological space obtained from $X_n$ by gluing on $(n+1)$-cells. For $n \in \mathbb{N}$ write $nCells \in Set$ for the set of $n$-cells of $X$.
The relative singular homology of the filtering degrees is
where $\mathbb{Z}[nCells]$ denotes the free abelian group on the set of $n$-cells.
The proof is spelled out at Relative singular homology - Of CW complexes.
With $k,n \in \mathbb{N}$ we have
In particular if $X$ is a CW-complex of finite dimension $dim X$ (the maximum degree of cells), then
Moreover, for $k \lt n$ the inclusion
is an isomorphism and for $k = n$ we have an isomorphism
This is mostly for instance in (Hatcher, lemma 2.34 b),c)).
By the long exact sequence in relative homology, discussed at Relative homology – long exact sequences, we have an exact sequence
Now by prop. 6 the leftmost and rightmost homology groups here vanish when $k \neq n$ and $k \neq n-1$ and hence exactness implies that
is an isomorphism for $k \neq n,n-1$. This implies the first claims by induction on $n$.
Finally for the last claim use that the above exact sequence gives
and hence that with the above the map $H_{n-1}(X_{n-1}) \to H_{n-1}(X)$ is surjective.
The geometric realization of any locally finite simplicial set is a CW-complex (Milnor 57).
any noncompact smooth manifold of dimension $n$ is homotopy equivalent to an $(n-1)$-dimensional CW-complex. (Napier-Ramachandran).
examples of universal constructions of topological spaces:
$\phantom{AAAA}$limits | $\phantom{AAAA}$colimits |
---|---|
$\,$ point space$\,$ | $\,$ empty space $\,$ |
$\,$ product topological space $\,$ | $\,$ disjoint union topological space $\,$ |
$\,$ topological subspace $\,$ | $\,$ quotient topological space $\,$ |
$\,$ fiber space $\,$ | $\,$ space attachment $\,$ |
$\,$ mapping cocylinder, mapping cocone $\,$ | $\,$ mapping cylinder, mapping cone, mapping telescope $\,$ |
$\,$ cell complex, CW-complex $\,$ |
Basic textbook accounts include
Brayton Gray, Homotopy Theory: An Introduction to Algebraic Topology, Academic Press, New York (1975).
George Whitehead, chapter II of Elements of homotopy theory, 1978
Peter May, A Concise Course in Algebraic Topology, U. Chicago Press (1999)
Allen Hatcher, Algebraic Topology, 2002
Allen Hatcher, Topology of cell complexes (pdf) in Algebraic Topology
Allen Hatcher, Vector bundles & K-theory (web)
Rudolf Fritsch and Renzo A. Piccinini, Cellular structures in topology, Cambridge studies in advanced mathematics Vol. 19, Cambridge University Press (1990). (pdf)
Marcelo Aguilar, Samuel Gitler, Carlos Prieto, section 5.1 of Algebraic topology from a homotopical viewpoint, Springer (2002) (toc pdf)
Original articles include
John Milnor, On spaces having the homotopy type of a CW-complex, Trans. Amer. Math. Soc. 90 (2) (1959), 272-280.
John Milnor, The geometric realization of a semi-simplicial complex, Annals of Mathematics, 2nd Ser., 65, n. 2. (Mar., 1957), pp. 357-362; pdf
See also
An inconclusive discussion here about what parts of the definition a CW complex should be properties and what parts should be structure.