Metrization of Weakly Developable Spaces

M artin (1976) introduced the concept of weak developability in order to study the problem of the metrization of spaces with weak bases. A space X is weakly developable if and only if there is a sequence { } n n N G ∈ of covers of X such that for each x X ∈ ,{ ( ) } , : n st x G n N ∈ is a weak base at x . The sequence { } n n N G ∈ is said to be a weak development for the space X . If each consists of open sets, then { n G } n n N G ∈ is a development for the space X and X is a developable space (Gruenhage, 1984). The idea of weak base was introduced by Arhangel'skii (1966) in the study of symmetrizable spaces. It is more convenient to use the form of Siwiec, (1974) and Franklin, (1965). A collection w of subsets of a space X is called a weak base for X provided that to each x X ∈ , there exists w such that x w ⊂ 1. Each member of contains x w x . 2. For any two members W and W 1 2 of , there is a W x w 3 in x w , such that W W 3 1 W ⊂ ∩ 2 . 3. A subset of F X is closed if and only if for every point x F ∉ , there exists a W in such that F W x w φ ∩ = . If to each x X ∈ we assign a collection x w of supersets of { } x such that { } : x x X = ∪ ∈ W w is a weak base by virtue of the collections , i.e., the collections x w x w


Introduction
M artin (1976) introduced the concept of weak developability in order to study the problem of the metrization of spaces with weak bases.A space X is weakly developable if and only if there is a sequence { } G ∈ is a development for the space X and X is a developable space (Gruenhage, 1984).The idea of weak base was introduced by Arhangel'skii (1966) in the study of symmetrizable spaces.It is more convenient to use the form of Siwiec, (1974) and Franklin, (1965).
A collection w of subsets of a space X is called a weak base for X provided that to each x X ∈ , there exists w such that x w ⊂ 1.Each member of contains x w x .2. For any two members W and W 1 2 of , there is a If to each x X ∈ we assign a collection x w of supersets of { } x such that { } is a weak base by virtue of the collections , i.e., the collections (3) of the preceding paragraph, then we say that the collection is a local weak base at w x for each x X ∈ .It is easy to show that a subset O of a space X with local weak bases { } open if and only if for each x O ∈ , there is a member W of the local weak base of In this study, we prove a metrization theorem for weakly developable spaces.We assume throughout this note that all spaces are T .A topological space is a T -space if, and only if, for each pair 0 0 x and of distint points, there is a nighborhood of one point to which the other does not y belong.Also, we let denote the set of all positive integers.For a collection G of subsets of a space N X , we define ( ) { }

lemma 1
Let { n N ∈ be a weak development of a space X .Then for every compact subset K of and any sequence of points in , there is a point in and a subsequence Proof.Let be a compact subset of K X and let be any sequence of points in .
Suppose there is no subsequence of : Then, we note that is a closed subset of X .For if it is not true then for some point we shall have y X F ∈ − ( ) for each i N ∈ .This will imply that the subsequence : n y i will converge to , which will contradict our assumption.Therefore, is closed.
It is easy to see that it does not contain a finite subcover of which contradicts the fact that is compact.Hence, the sequences K K : n y n N have a convergent subsequence.

lemma 2
Let { n N ∈ be a weak development of a space X , which satisfies the following condition: For any closed subset F of X and any point , Then every compact subset of X is closed.
Proof.Let K be a compact subset of X .Suppose is not closed.Then there is a point such that for all i .Thus for each i , let .Hence, the sequence This is not possible since a weakly developable space is T and hence by the hypothesis there is an is an open cover of with no finite subcover of giving a contradiction.K K ∩

Theorem
The following are equivalent for a space X .
1.The space X is metrizable.
2. The space X has a weak development { } n n N G ∈ such that for any closed subset of X and any point , X F ∈ − there is an i such that N ∈ ( ) ( ) x .The sequence { } n n N G ∈ is said to be a weak development for the space X .If each consists of open sets, then {