see also algebraic topology, functional analysis and homotopy theory
Basic concepts
topological space (see also locale)
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
subsets are closed in a closed subspace precisely if they are closed in the ambient space
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
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
Basic homotopy theory
A partition of unity is a partition of the unit function on a topological space into a sum of continuous functions that are each non-zero only on small parts of the space.
Let $X$ be a topological space. A (point finite) partition of unity on $X$ is a collection $\{u_j\}_{j \in J}$ of continuous functions $u_j \colon X \to [0,1]$, $j\in J$ to the closed interval with its Euclidean metric topology such that
For each $x\in X$, there is only a finite number of $j\in J$ such that $u_j(x) \neq 0$ (point finiteness condition);
$\sum_{j \in J} u_j(x) = 1$ for all $x\in X$.
A partition of unity defines an open cover of $X$, consisting of the open sets $u_j^{-1}(0,1]$. Call this the induced cover.
Sometimes (rarely) the condition that $\{u_j\}_J$ is point finite is dropped. In this case we refer to a non-point finite partition of unity (see red herring principle). In this case for each point of $X$ at most countably-many of the functions $u_j$ are non-zero, and we have to interpret the sum in 1. above as being a convergent infinite series.
Given a cover $\mathcal{U} = \{U_j\}_{j\in J}$ of a topological space (open cover or closed or neither), the partition of unity $\{u_j\}_J$ is subordinate to $\mathcal{U}$ if for all $j\in J$,
What this means is that the open sets $u_j^{-1}(0,1]$ form an open cover refining the cover $\mathcal{U}$.
(paracompact Hausdorff spaces equivalently admit subordinate partitions of unity)
Assuming the axiom of choice then:
Let $(X,\tau)$ be a topological space. Then the following are equivalent:
$(X,\tau)$ is a paracompact Hausdorff space.
Every open cover of $(X,\tau)$ admits a subordinate partition of unity.
Similarly normal spaces are equivalently those such that every locally finite cover has a subordinate partition of unity (reference Bourbaki, Topology Generale - find this!)
Paracompact smooth manifolds even have smooth partitions of unity subordinate to any open cover (this follows from the existence of a smooth bump function on $[-1,1]$). It is not true, however, that analytic manifolds have analytic partitions of unity - the aforementioned bump function is smooth but not analytic:
(smooth manifolds admit smooth partitions of unity)
Let $X$ be a paracompact smooth manifold. Then every open cover $\{U_i \subset X\}_{i \in I}$ has a subordinate partition of unity by functions $\{f_i \colon U_i \to \mathbb{R}\}_{i \in I}$ which are smooth functions.
Since $X$ is paracompact, the given open cover has a refinement to a locally finite cover, and then there exists one with the same index set (this prop.):
Recall that the smooth manifold $X$ is a normal topological space, because it is a paracompact Hausdorff space by definition and paracompact Hausdorff spaces are normal. Therefore we may invoke the shrinking lemma to obtain yet another open cover with the same index set
with the property that
Now let
be an atlas exhibiting the smooth structure on the smooth manifold. Then by definition, for each point $x \in X$ there is $i_x \in I$ and $j_x \in J$ such that
By the nature of the subspace topology, this intersection is still an open subset of $Im(\phi_{j_x}) \simeq \mathbb{R}^n$. Therefore by the definition of the metric topology there exists a positive real number $\epsilon_{x}$ such that the open ball of radius $\epsilon_x$ around $x$ is an open neighbourhood of $x$ still contained in $V_{i_x}$:
Let then
be the collection of choices of such open balls, around each point of the manifold. This is an open cover which refines the cover $\{U'_i \subset X\}_{i \in I}$. Again by paracompactness of $X$, there exists a locally finite subcover, hence a subset of points $S \subset X$ such that
is a locally finite open cover of $X$.
Let then
a set of smooth bump functions whose support is the topological closure of the chosen open ball around $s$, regarded now as a subspace of the corresponding $j$-th copy of $\mathbb{R}^n$:
Hence the smooth bump functions $b_s$ vanish on $\mathbb{R}^n \backslash Cl(B^\circ_s(\epsilon_s))$, such that their extension by zero to functions $\hat b_s$ on all of $X$
are still smooth functions: $\hat b_s \in C^\infty(X,\mathbb{R})$.
Now by local finiteness of both the cover $\{U'_i \subset X\}_{i \in I}$ and of the cover $\{B^\circ_s(\epsilon_s)\}_{s \in S}$ we have that the sum
is well defined for each $x \in X$ (only finitely many of the summands are non-zero) and by the covering property each point $x$ is contained in at least one of the patches of the cover, hence in the interior of the support of at least one of the $\hat b_s$ and so
for all $x \in X$. This means that it makes sense to define
and these are still smooth functions: $f_i \in C^\infty(X,\mathbb{R})$.
We claim now that these form the required partition of unity subordinate to the original cover:
By construction of the various open covers we have
and hence
By construction of the functions $f_i$ we have
A collection of functions $\mathcal{U} = \{u_i : X \to [0,1]\}$ such that every $x\in X$ is in the support of some $u_i$. Then $\mathcal{U}$ is called locally finite if the cover $u_i^{-1}(0,1]$ (i.e. the induced cover) is locally finite.
(Mather, 1965)
Let $\{u_i\}_J$ be a non-point finite partition of unity. Then there is a locally finite partition of unity $\{v_i\}_{i\in J}$ such that the induced cover of the latter is a refinement of the induced cover of the former.
(For a proof, see p.354 of Dold’s Lectures on algebraic topology. Google books link to page 354, which may or may not be visible)
This implies that (loc. finite) numerable covers are cofinal in induced covers arising from collections of functions as in the definition. In particular, given the Milnor classifying space $\mathcal{B}^M G$ of a topological group $G$, which comes with a countable family of ‘coordinate functions’ $\mathcal{B}^M G \to [0,1]$, has a numerable cover. This is shown by Dold to be a trivialising cover for the universal bundle constructed by Milnor, and so the universal bundle is numerable?.
Partitions of unity can be used in constructing maps from spaces to geometric realizations of simplicial spaces (incl. simplicial sets) - for example a classifying map for a $G$-bundle where $G$ is a Lie group.
Partitions of unity can be used to give explicit coboundaries for the cocycles of the complex of functions on a cover.
Let $\{U_i \to X\}$ be a open cover and $\{\rho_i \in C(X,\mathbb{R})\}$ a collection of functions with
$(x not \in U_i) \Rightarrow \rho_i(x) = 0$
$\sum_i \rho_i = const_1$.
Write $C(\{U_i\}) : \Delta^{op} \to Top$ for the Cech nerve of the cover and $C(C(\{U_i\}), \mathbb{R})$ for the cosimplicial ring of functions on this simplicial topological space; and $(C_\bullet(C(\{U_i\}), \mathbb{R}), \delta)$ for the corresponding (normalized) cochain complex: its differential is the alternating sum of the pullbacks of functions along the face maps, i.e. along the restriction maps
For instance for $f = \{f_{i_1, i_2, \cdots, i_n} \in C(U_{i_1} \cap \cdots \cap U_{i_{n+1}})\}$ a collection of functions in degree $n$, we have
This cochain complex has vanishing cochain cohomology in positive degree. We can explicitly construct corresponding coboundaries using the partion of unity:
assume that with the above notation $f$ is a cocycle in positive degree, in that $\delta f = 0$. Then define the $(n-1)$-cochain
Here in the summands on the right the product is defined on $U_{i_0} \cap U_{i_1} \cap \cdots \cap U_{i_n}$ and extended as 0 to all of $U_{i_1} \cap \cdots \cap U_{i_n}$.
With this definition we have
To see this we compute
where in the second step we used the condition $\delta f = 0$ and in the last step we used the property of the partition of unity.
This construction is used a lot in Cech cohomology. For instance it can be used to show in Chech cocycles that every principal bundle admits a connection on a bundle (see there for the details).
Albrecht Dold, Partitions of unity in the theory of fibrations, Ann. of Math. 78. (1963), 223-255.
Albrecht Dold, Lectures on algebraic topology, Springer Classics in Mathematics (1980), p.354.
M. Mather, Paracompactness and partitions of unity, PhD thesis, Cambridge (1965).
Discussion of partitions of unity in constructive mathematics is in