Refinement

generalized development

A generalized development is a collection $F$ of families of subsets of a topological space $X$ such that for every $x\in X$ and every neighbourhood $O_x$ of $x$ there exists a $\gamma\in F$ such that the union of all the elements of the family $\gamma$ containing $x$ (the so-called star $\operatorname{St}_\gamma(x)$ of $x$ relative to $\gamma$) is contained in $O_x$.

Generalized developments consisting of open coverings are important. They play an essential role in dimension theory, the theory of compactifications, the theory of uniform spaces, the theory of continuous mappings, and in metrization problems. Informally, for a collection of open coverings to be a generalized development means that this collection approximates the given space near every point. Often one requires specific relations between the families in a generalized development — e.g., if one requires that: a) for every family in the collection there is another family in the collection that is a star-refinement of it; and b) for every two families in the collection there is a third family that refines both, then one obtains the definition of a (base for a) uniform structure compatible with the given topology.

Generalized developments consisting of locally finite coverings are considered in connection with the theory of paracompact spaces, and generalized developments consisting of finite open coverings in connection with the theory of compact spaces. In dimension theory, generalized developments consisting of open coverings of given multiplicity directed by the relation of "being a refinement of" have special meaning. Generalized developments consisting of closed coverings on which no restrictions such as local finiteness have been imposed are of no interest; for example, the covering of a $T_1$-space by its singleton subsets forms by itself a generalized development that carries no information on the topology of the space.

Countable generalized developments consisting of open coverings (simply called developments) play an important role; they are often written down by arbitrarily listing their members by the natural numbers in such a way that every covering is refined by the next one in the sequence. Developments are introduced at the first stage in the problem of metrizability of spaces, since their existence is a necessary condition for metrizability. This condition is not sufficient in the class of completely-regular spaces, but the addition of paracompactness (which is a consequence of metrizability) makes it sufficient. More exactly, a $T_1$-space is metrizable if and only if it is collectionwise normal and has a development. In particular, a compactum with a development is metrizable. It is not known (1978) whether there exists a non-metrizable normal space with a development, without further axiomatic assumptions; however, it is known that the existence of such a space is compatible with the Zermelo–Fraenkel axioms, although no "naive" example has been constructed so far.

The class of spaces with developments has good properties. It is closed under the operations of taking subspaces and countable products, and it is stable under perfect mappings. However, a whole series of regularities holding in the class of metrizable spaces do not hold for spaces with developments. Thus, a separable space with a development need not have a countable base. A space with a development is paracompact if and only if it is metrizable. Though not generally metrizable, spaces with developments allow of a sort of generalized metrization by means of écarts satisfying Cauchy's condition. There is also a convenient characterization of spaces with developments as images of metric spaces under continuous mappings subject to the requirement: the pre-image of every point is at a positive distance from the complement of the pre-image of every neighbourhood of this point.

References

[1]	A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian)

Comments

The notion defined in the article above is never called "refinement" in the West (see below). In fact, the terminology "generalized development" is also seldom used. A generally used notion is that of "development" .

The problem whether every normal space with a development, the so-called normal Moore space problem (a regular space with a development is called a Moore space), is solved now. In 1978 P. Nyikos showed that, assuming the product measure extension axiom (PMEA), every normal Moore space is metrizable. To show that PMEA is consistent with the usual axioms of set theory one needs to assume the existence of large cardinals. In 1983 W.G. Fleissner completed the solution by showing that if every normal Moore space is metrizable, then one can show that the existence of measurable cardinals is consistent with the usual axioms of set theory, see [a1].

Refinements.

A collection $\mathcal F$ of subsets of a set $X$ is called a refinement of a collection $\mathcal G$ if for every $F\in\mathcal F$ there is a $G\in\mathcal G$ such that $F\subset G$.

The most common situation is when $X$ is a topological space, $\mathcal G$ is an open covering of $X$ and $\mathcal F$ is also a covering of $X$ (indeed, if $\bigcup\mathcal F\neq X$, then $\mathcal F$ is often called a partial refinement of $\mathcal G$).

By requiring that every open covering has a refinement of a particular kind, one obtains various interesting classes of spaces, best known of which is probably the class of paracompact spaces: A space is defined to be paracompact if every open covering of it has a locally finite open refinement.

In dimension theory one defines a normal space to have covering dimension at most $n$ if every finite open covering has a finite open refinement such that every point is in at most $n+1$ elements of the refinement.

References

[a1]	W.G. Fleissner, "The normal Moore space conjecture and large cardinals" K. Kunen (ed.) J.E. Vaughan (ed.) , Handbook of Set-Theoretic Topology , North-Holland (1984) pp. 733–760