The concept of network is a useful tool in working with generalized metric spaces. A network is like a base for a topology, but the members of a network do not have to be open. After a brief discussion on network, the focus here is on the spaces with networks that are countably infinite in size. The following facts are presented:
- Any space with a countable network is separable and Lindelof.
- The property of having a countable network is hereditary. Thus any space with a countable network is hereditarily separable and hereditarily Lindelof.
- The property of having a countable network is preserved by taking countable product.
- The Sorgenfrey Line is an example of a hereditarily separable and hereditarily Lindelof space that has no countable network.
- For any compact space , . In particular, any compact space with a countable network is metrizable.
- As a corollary to 5, for any compact .
- A space has a countable network if and only if it is the continuous impage of a separable metric space (hence such a space is sometimes called cosmic).
- Any continuous image of a cosmic space is cosmic.
- Any continuous image of a compact metric space is a compact metric space.
- As a corollary to 2, any space with countable network is perfectly normal.
- An example is given to show that the continuous image of a separable metric space needs not be metric (i.e. an example of a cosmic space that is not metrizable).
All spaces in this discussion are at least (Hausdorff and regular). Let be a space. A collection of subsets of is said to be a network for if for each and for each open with , then we have for some . The network weight of a space , denoted by , is defined as the minimum cardinality of all the possible where is a network for . The weight of a space , denoted by , is defined as the minimum cardinality of all possible where is a base for . Obviously any base is also a network. Thus . For any compact space , . On the other hand, the set of singleton sets is a network. Thus .
Our discussion is based on an important observation. Let be the topology for the space . Let . We can find a base that generates a weaker (coarser) topology such that . We can also find a base that generates a finer topology such that . These are restated as lemmas.
Lemma 1. We can define base that generates a weaker (coarser) topology on such that . Thus .
Proof. Let be a network for such that . Consider all pairs such that there exist disjoint with and . Such pairs exist because we are working in a Hausdorff space. Let be the collection of all such open sets and their finite interections. This is a base for a topology and let be the topology generated by . Clearly, and this is a Hausdorff topology. Note that .
Lemma 2. We can define base that generates a finer topology on such that . Thus .
Proof. As before, let be a network for such that . Since we are working in a regular space, we can assume that the sets in are closed. If not, take closures of the elements of and we still have a network. Consider to be the set of all finite intersections of elements in . This is a base for a topology on . Let be the topology generated by this base. Clearly, . It is also clear that . The only thing left to show is that the finer topology is regular. Note that the network consists of closed sets in the topology . Thus the sets in the base also consists of closed sets with respect to and the sets in are thus closed in the finer topology. Since is a base consisting of cloased and open sets, the topology regular.
Discussion of 1, 2, and 3
Points 1, 2 and 3 are basic facts about countable network and they are easily verified based on definitions. They are called out for the sake of having a record.
Discussion of 4
The Sorgenfrey Line does not have a countable network for the same reason that the Sorgenfrey Plane is not Lindelof. If the Sorgenfrey Line has a countable netowrk, then the Sorgenfrey plane would have a countable network and hence Lindelof.
Discussion of 5
In general, . In a compact Hausdorff space, any weaker Hausdorff topology must conincide with the original topology. So the weaker topology produced in Lemma 1 must coincide with the original topology. In the countable case, any compact space with a countable network has a weaker topology with a countable base. This weaker topology must coincide with the original topology.
Discussion of 6
Note that always holds. For compact spaces, we have .
Discussion of 7
Let be a space with a countable network. By Lemma 2, has a finer topology that has a countable base. Let denote with this finer second countable topology. Then the identity map from onto is continuous.
For the other direction, let be a continuous function mapping a separable metric space onto . Let be a countable base for . Then is a network for .
Discussion of 8
This is easily verified. Let is the continuous image of a cosmic space . Then is the continuous image of some separable metric space . It follows that is the continuous image of .
Discussion of 9
Let be compact metrizable and let be a continuous image of . Then is compact. By point 7, has a countable network. By point 5, is metrizable.
Discussion of 10
A space is perfectly normal if it is normal and that every closed subset is a set. Let be a space with a countable network. The normality of comes from the fact that it is regular and Lindelof. Note that is also hereditarily Lindelof. In a hereditarily Lindelof and regular space, every open subspace is an set (thus every closed set is a set.
Discussion of 11 (Example of cosmic but not separable metrizable space)
This is the “Butterfly” space or “Bow-tie” space due to L. F. McAuley. I found this example in [Michael]. Let where
Points in have the usual plane open neighborhoods. A basic open set at is of the form where consists of and all points having distance from and lying underneath either one of the two straight lines in which emanate from and have slopes and , respectively.
It is clear that is a Hausdorff and regular space. The relative “Bow-tie” topologies on and coincide with the usual topology on and , respectively. Thus the union of the usual countable bases on and would be a countable network for . On the other hand, is separable but cannot have a countable base (hence not metrizable).
Michael, E., spaces, J. Math. Mech. 15, 983-1002.