CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
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 $\{u_j\}_J$ of continuous functions $u_j:X \to [0,1]$, $j\in J$ such that
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).
Given a cover $\mathcal{U} = \{U_j\}_{j\in J}$ of a topological space (open 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 spaces have the property that every open cover has a subordinate partition of unity.
Normal spaces have the property that every locally finite cover has a subordinate partition of unity.
The last two are actually characterisations of paracompact resp. normal spaces (reference Bourbaki, Topology Generale - find this!)
Paracompact smooth manifolds 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.
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.)
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).