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
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 $\mathcal{U}$.
Consider $\mathbb{R}$ with its Euclidean metric topology.
Let $\epsilon \in (0,\infty)$ and consider the open cover
Then a partition of unity $\{ f_n \colon \mathbb{R} \to [0,1] \}_{n \in \mathbb{N}}$ subordinate to this cover is given by
(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:
(open cover of smooth manifold admits locally finite refinement by closed balls)
Let $X$ be a smooth manifold and let $\{U_i \subset X\}_{i \in I}$ be an open cover. Then there exists cover
which is a locally finite refinement of $\{U_i \subset X\}_{i \in I}$ with each patch diffeomorphic to a closed ball in Euclidean space.
First consider the special case that $X$ is compact topological space.
Let
be a smooth atlas representing the smooth structure on $X$. The intersections
still form an open cover of $X$. Hence for each point $x \in X$ there is $i \in I$ and $j \in J$ with $x \in U_i \cap V_j$. By the nature of the Euclidean topology, there exists a closed ball $B_x$ around $\phi_j^{-1}(x)$ in $\phi_j^{-1}(U_i \cap V_j) \subset \mathbb{R}^n$. Its image $\phi_j(B_x) \subset X$ is a neighbourhood of $x \in X$ diffeomorphic to a closed ball.
The interiors of these balls form an open cover
of $X$ which, by construction, is a refinement of $\{U_i \subset X\}_{i \in I}$. By the assumption that $X$ is compact, this has a finite subcover
for $L$ a finite set. Hence
is a finite cover by closed balls, hence in particular locally finite, and by construction it is still a refinement of the orignal cover. This shows the statement for $X$ compact.
Now for general $X$, notice that without restriction we may assume that $X$ is connected, for if it is not, then we obtain the required refinement on all of $X$ by finding one on each connected component.
But if a locally Euclidean paracompact Hausdorff space $X$ is connected, then it is sigma-compact and in fact admits a countable increasing exhaustion
by open subsets whose topological closures
exhaust $X$ by compact subspaces $K_n$ (by the proof of this prop.).
For $n \in \mathbb{N}$, consider the open subspace
which canonically inherits the structure of a smooth manifold (this prop.). As above we find a refinement of the restriction of $\{U_i \subset X\}_{i \in I}$ to this open subset by closed balls and since the further subspace $K_{n+1}\setminus K_n$ is still compact (by this lemma) there is a finite set $L_n$ such that
is a finite cover of $K_{n+1} \setminus K_n$ by closed balls refining the original cover.
It follows that the union of all these
is a refinement by closed balls as required. Its local finiteness follows by the fact that each $B_{l_n}$ is contained in the “strip” $V_{n+2} \setminus K_{n-1}$, each strip contains only a finite set of $B_{l_n}$-s and each strip intersects only a finite number of other strips. (Hence an open subset around a point $x$ which intersects only a finite number of elements of the refined cover is given by any one of the balls $B_{l_n}$ that contain $x$.)
(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.
By lemma 1 the given cover has a locally finite refinement by closed subsets diffeomorphic to closed balls:
Given this, let
be the function which on $V_j$ is given by a smooth bump function
with support $supp(b_j) = B_0(\epsilon_j)$:
By the nature of bump functions this is indeed a smooth function on all of $X$. By local finiteness of the cover by closed balls, the function
given by
is well defined (the sum involves only a finite number of non-vanishing contributions) and is smooth. Therefore setting
then
is a subordinate partition of unity by smooth functions as required.
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