Paracompact space

A topological space in which any open covering (cf. Covering (of a set)) can be refined to a locally finite open covering. (A collection $\gamma$ of subsets of a topological space $X$ is called locally finite in $X$ if each point $x\in X$ has a neighbourhood in $X$ which intersects only finitely many elements of the collection $\gamma$; a collection $\gamma$ of subsets is a refinement of a collection $\lambda$ if each element of the collection $\gamma$ is contained in some element of the collection $\lambda$.) A paracompact Hausdorff space is called a paracompactum. The class of paracompacta is very extensive — it includes all metric spaces (Stone's theorem) and all Hausdorff compacta. However, not every locally compact Hausdorff space is paracompact.
The significance of paracompactness is shown by the clear generality of this concept and by a number of remarkable properties of paracompacta. First of all, each paracompact Hausdorff space is normal (cf. Normal space). This permits the construction in paracompacta of a partition of unity subordinate to an arbitrary given open covering $\gamma$. That is, a collection of real non-negative continuous functions on the space subject to the following conditions: a) the collection of supports of these functions is locally finite and is a refinement of $\gamma$; and b) at each point of the space, the sum of the values, at that point, of all those functions of the set which are different at it from zero (there are only finitely many such functions) is equal to one. Partitions of unity are basic tools for the construction of immersions of spaces in standard spaces. In particular, they are used for imbedding manifolds in Euclidean spaces and for the proof of the theorem about the metrizability of each Tikhonov space with a $\sigma$-locally finite base. Moreover, in the theory of manifolds partitions of unity form the basis of methods with the help of which a simultaneous synthesis of local constructions is obtained, after first constructing within the boundaries of individual charts objects with certain desired properties (in particular, for vector or tensor fields). Therefore one of the requirements assumed in the theory of manifolds is the requirement of paracompactness, which is not superfluous since there exist non-paracompact connected Hausdorff spaces, which are locally like $\mathbf R^n$.
In the class of paracompacta the metrizability conditions are simplified. In particular, a paracompactum is metrizable if and only if it possesses a base of countable order, i.e. a base such that any decreasing sequence of elements of it containing an arbitrary point $x\in X$ forms a base at that point. There are many paracompactness criteria. In particular, for a Tikhonov space $X$ the following conditions are equivalent: a) $X$ is paracompact; b) any open covering of $X$ can be refined to a locally finite covering; c) any open covering of $X$ can be refined to a $\sigma$-locally finite open covering; and d) any open covering of $X$ can be refined to a conservative closed covering, i.e. to a covering such that the union of any subcollection of it is closed in $X$.
The following criterion is important: A Tikhonov space is paracompact if and only if each open covering $\gamma$ of it has an open star refinement $\lambda$; the latter means that for each point $x\in X$ the union of all elements of $\lambda$ containing $x$ is contained in some element of $\gamma$. The concept of star refinability serves as an expression of the idea of unlimited division of a space, and can be interpreted as the most general set-theoretical form of the triangle axiom.