Locally finite covering

A covering (cf. Covering (of a set)) of a topological space by subsets of it such that every point has a neighbourhood that intersects only finitely many elements of this covering. One cannot select a locally finite covering from every open covering of a straight line: it is sufficient to consider a monotone sequence of intervals that increase in length without limit. It turns out that the possibility of selecting a locally finite covering from any open covering of a space is equivalent to compactness of the space. The idea of local finiteness in conjunction with the concept of refinement carries an essentially new meaning. A.H. Stone's theorem asserts that any open covering of an arbitrary metric space can be refined to a locally finite covering. Hausdorff spaces that have the latter property are said to be paracompact (cf. Paracompact space). Locally finite coverings are important not only because of their participation in the definition of paracompactness. The requirement of local finiteness plays an essential role in constructions belonging to dimension theory and in the statements and proofs of addition theorems of various kinds. The existence in a regular space of a base that splits into a union of a countable family of locally finite open coverings is equivalent to the metrizability of this space. Open locally finite coverings of a normal space serve as a construction of a partition of unity on this space, subordinate to this covering. By means of partitions of unity it has been possible to construct, in particular, standard mappings of manifolds into Euclidean spaces. The requirement of local finiteness of a covering is not necessarily connected with the assumption that it is open. Local finiteness of a covering of a space automatically implies that in this covering there are "sufficiently many" sets that are close in their properties to open sets. If any open covering of a regular space can be refined to a locally finite covering, that space is paracompact. Locally finite families of sets in a space, defined similarly but not necessarily covering the space, have also been considered. A special case of them are discrete families of sets: families of sets such that each point in the whole space has a neighbourhood that intersects at most one element of this family. Discrete families are important in connection with the study of separation in a space. Thus, collectively-normal spaces are distinguished by the requirement that any discrete family of sets is separated by a discrete family of neighbourhoods. This condition is directly connected with the problem of the combinatorial extension of locally finite families of sets to locally finite families of open sets.

References

[1]	R. Engelking, "General topology" , PWN (1977)
[2]	A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian)
[3]	P. [P.S. Aleksandrov] Alexandroff, "Sur les ensembles de la première classe et les espaces abstraits" C.R. Acad. Sci. Paris , 178 (1924) pp. 185–187
[4]	A.H. Stone, "Paracompactness and product spaces" Bull. Amer. Math. Soc. , 54 (1948) pp. 977–982
[5]	E.A. Michael, "A note on paracompact spaces" Proc. Amer. Math. Soc. , 4 (1953) pp. 831–838

Comments

A partition of unity on a space $ X $ is a family of continuous functions $ \{ f _ {i} \} _ {i} $ from $ X $ to $ [ 0 , 1 ] $ such that $ \sum _ {i} f _ {i} ( x) = 1 $ for all $ x \in X $. It is said to be subordinate to a covering $ \mathfrak U $ if the open covering $ \{ f _ {i} ^ { - 1 } [ ( 0 , 1 ] ] \} _ {i} $ refines $ \mathfrak U $.

A (locally finite) family $ \{ U _ {i} \} _ {i} $ of (open) sets is a combinatorial extension of a (locally finite) family $ \{ F _ {i} \} _ {i} $ of sets if $ F _ {i} \subseteq U _ {i} $ for all $ i $ and for every set $ I $ of indices $ \cap _ {i \in I } F _ {i} = \emptyset $ implies $ \cap _ {i \in I } U _ {i} = \emptyset $.

References

[a1]	D.K. Burke, "Covering properties" K. Kunen (ed.) J.E. Vaughan (ed.) , Handbook of Set-Theoretic Topology , North-Holland (1984) pp. Chapt. 9; pp. 347–422