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 be a topological space. A partition of unity on is a collection of continuous functions , to the closed interval with its Euclidean metric topology such that for all .
A partition of unity defines an open cover of , consisting of the open sets . Call this the induced cover.
Given a cover of a topological space (open cover or closed or neither), the partition of unity is subordinate to if for all ,
What this means is that the open sets form an open cover refining the cover .
A partition of unity is point finite if for every there is only a finite number of such that .
A partition of unity is locally finite if for every there is an open neighborhood of such that for only a finite number of there is such that .
Often, the property of local finiteness is included in the definition of a partition of unity. This is harmless, since a result due to Michael R. Mather (Prop. below) says that for any partition of unity we can find a locally finite partition of unity with the same indexing set and whose induced cover refines the original induced cover.
Consider with its Euclidean metric topology.
Let and consider the open cover
Then a partition of unity subordinate to this cover is given by
(paracompact Hausdorff spaces equivalently admit subordinate partitions of unity)
Assuming the axiom of choice then:
Let be a topological space. Then the following are equivalent:
Every open cover of 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!)
Slightly more generally, a topological space (not necessarily Hausdorff) is fully normal if and only every open cover admits a subordinate partition of unity.
A T1-space is fully normal if and only if it is paracompact, in which case it is also Hausdorff.
For topological spaces that are not T1-spaces, the condition of being fully normal is strictly stronger than paracompactness.
A regular locale is fully normal if and only if it is paracompact.
The usual proof of the existence of partitions of unity goes through for such locales since it does not make any use of points.
Paracompact smooth manifolds admit locally finite smooth partitions of unity subordinate to any open cover (this follows from the existence of a smooth bump function on ). 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 be a smooth manifold and let be an open cover. Then there exists cover
which is a locally finite refinement of with each patch diffeomorphic to a closed ball in Euclidean space.
First consider the special case that is compact topological space.
Let
be a smooth atlas representing the smooth structure on . The intersections
still form an open cover of . Hence for each point there is and with . By the nature of the Euclidean topology, there exists a closed ball around in . Its image is a neighbourhood of diffeomorphic to a closed ball.
The interiors of these balls form an open cover
of which, by construction, is a refinement of . By the assumption that is compact, this has a finite subcover
for 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 compact.
Now for general , notice that without restriction we may assume that is connected, for if it is not, then we obtain the required refinement on all of by finding one on each connected component.
But if a locally Euclidean paracompact Hausdorff space is connected, then it is sigma-compact and in fact admits a countable increasing exhaustion
by open subsets whose topological closures
exhaust by compact subspaces (by the proof of this prop.).
For , 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 to this open subset by closed balls and since the further subspace is still compact (by this lemma) there is a finite set such that
is a finite cover of 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 is contained in the “strip” , each strip contains only a finite set of -s and each strip intersects only a finite number of other strips. (Hence an open subset around a point which intersects only a finite number of elements of the refined cover is given by any one of the balls that contain .)
(smooth manifolds admit locally finite smooth partitions of unity)
Let be a paracompact smooth manifold. Then every open cover has a subordinate partition of unity by functions which are smooth functions.
By lemma the given cover has a locally finite refinement by closed subsets diffeomorphic to closed balls:
Given this, let
be the function which on is given by a smooth bump function
with support :
By the nature of bump functions this is indeed a smooth function on all of . 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 such that every is in the support of some . Then is called locally finite if the cover (i.e. the induced cover) is locally finite.
Let be a partition of unity. Then there is a locally finite partition of unity such that the induced cover of the latter is a refinement of the induced cover of the former.
The proof is Proposition A.2.8 in Dold 95 and Lemma 5.1.8 on page 301 of Engelking 89.
This implies that (locally finite) numerable covers are cofinal in induced covers arising from collections of functions as in the definition. In particular, given the Milnor classifying space of a topological group , which comes with a countable family of ‘coordinate functions’ , 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 -bundle where 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 be a open cover and a collection of functions with
.
Write for the Cech nerve of the cover and for the cosimplicial ring of functions on this simplicial topological space; and 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 a collection of functions in degree , 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 is a cocycle in positive degree, in that . Then define the -cochain
Here in the summands on the right the product is defined on and extended as 0 to all of .
With this definition we have
To see this we compute
where in the second step we used the condition 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.
Michael R. Mather, Paracompactness and partitions of unity, Mimeographed notes, Cambridge (1964).
Michael R. Mather, Products in spectral sequences and other topics, PhD dissertation, Cambridge (1966).
Albrecht Dold, p. 354 in: Lectures on Algebraic Topology, Springer 1995 (doi:10.1007/978-3-642-67821-9, pdf, GoogleBooks)
Ryszard Engelking, p. 301 in: General Topology, Sigma series in pure mathematics 6, Heldermann 1989 (ISBN 388538-006-4)
Discussion of partitions of unity in constructive mathematics is in
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