Spaces With Countable Network

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:

  1. Any space with a countable network is separable and Lindelof.
  2. The property of having a countable network is hereditary. Thus any space with a countable network is hereditarily separable and hereditarily Lindelof.
  3. The property of having a countable network is preserved by taking countable product.
  4. The Sorgenfrey Line is an example of a hereditarily separable and hereditarily Lindelof space that has no countable network.
  5. For any compact space X, nw(X)=w(X). In particular, any compact space with a countable network is metrizable.
  6. As a corollary to 5, w(X) \leq \vert X \vert for any compact X.
  7. A space X 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).
  8. Any continuous image of a cosmic space is cosmic.
  9. Any continuous image of a compact metric space is a compact metric space.
  10. As a corollary to 2, any space with countable network is perfectly normal.
  11. 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 T_3 (Hausdorff and regular). Let X be a space. A collection \mathcal{N} of subsets of X is said to be a network for X if for each x \in X and for each open U \subset X with x \in U, then we have x \in N \subset U for some N \in \mathcal{N}. The network weight of a space X, denoted by nw(X), is defined as the minimum cardinality of all the possible \vert \mathcal{N} \vert where \mathcal{N} is a network for X. The weight of a space X, denoted by w(X), is defined as the minimum cardinality of all possible \vert \mathcal{B} \vert where \mathcal{B} is a base for X. Obviously any base is also a network. Thus nw(X) \leq w(X). For any compact space X, nw(X)=w(X). On the other hand, the set of singleton sets is a network. Thus nw(X) \leq \vert X \vert.

Our discussion is based on an important observation. Let \mathcal{T} be the topology for the space X. Let \mathcal{K}=nw(X). We can find a base \mathcal{B}_0 that generates a weaker (coarser) topology such that \lvert \mathcal{B}_0 \lvert=\mathcal{K}. We can also find a base \mathcal{B}_1 that generates a finer topology such that \lvert \mathcal{B}_1 \lvert=\mathcal{K}. These are restated as lemmas.

Lemma 1. We can define base \mathcal{B}_0 that generates a weaker (coarser) topology \mathcal{S}_0 on X such that \lvert \mathcal{B}_0 \lvert=\mathcal{K}. Thus w(X,\mathcal{S}_0) \leq nw(X).

Proof. Let \mathcal{N} be a network for (X,\mathcal{T}) such that \vert \mathcal{N} \vert=nw(X,\mathcal{T}). Consider all pairs N_0,N_1 \in \mathcal{N} such that there exist disjoint O_0,O_1 \in \mathcal{T} with N_0 \subset O_0 and N_1 \subset O_1. Such pairs exist because we are working in a Hausdorff space. Let \mathcal{B}_0 be the collection of all such open sets O_0,O_1 and their finite interections. This is a base for a topology and let \mathcal{S}_0 be the topology generated by \mathcal{B}_0. Clearly, \mathcal{S}_0 \subset \mathcal{T} and this is a Hausdorff topology. Note that w(X,\mathcal{S}_0) \leq \vert \mathcal{B}_0 \vert =\vert \mathcal{N} \vert.

Lemma 2. We can define base \mathcal{B}_1 that generates a finer topology \mathcal{S}_1 on X such that \lvert \mathcal{B}_1 \lvert=\mathcal{K}. Thus w(X,\mathcal{S}_1) \leq nw(X).

Proof. As before, let \mathcal{N} be a network for (X,\mathcal{T}) such that \vert \mathcal{N} \vert=nw(X,\mathcal{T}). Since we are working in a regular space, we can assume that the sets in \mathcal{N} are closed. If not, take closures of the elements of \mathcal{N} and we still have a network. Consider \mathcal{B}_1 to be the set of all finite intersections of elements in \mathcal{N}. This is a base for a topology on X. Let \mathcal{S}_1 be the topology generated by this base. Clearly, \mathcal{T} \subset \mathcal{S}_1. It is also clear that w(X,\mathcal{S}_1) \leq nw(X). The only thing left to show is that the finer topology is regular. Note that the network \mathcal{N} consists of closed sets in the topology \mathcal{T}. Thus the sets in the base \mathcal{B}_1 also consists of closed sets with respect to \mathcal{T} and the sets in \mathcal{B}_1 are thus closed in the finer topology. Since \mathcal{B}_1 is a base consisting of cloased and open sets, the topology \mathcal{S}_1 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, nw(X) \leq w(X). 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 nw(X) \leq \lvert X \lvert always holds. For compact spaces, we have w(X)=nw(X) \leq \lvert X \lvert.

Discussion of 7
Let X be a space with a countable network. By Lemma 2, X has a finer topology that has a countable base. Let Y denote X with this finer second countable topology. Then the identity map from Y onto X is continuous.

For the other direction, let f:Y \rightarrow X be a continuous function mapping a separable metric space Y onto X. Let \mathcal{B} be a countable base for Y. Then \lbrace{f(B):B \in \mathcal{B}}\rbrace is a network for X.

Discussion of 8
This is easily verified. Let X is the continuous image of a cosmic space Y. Then Y is the continuous image of some separable metric space Z. It follows that X is the continuous image of Z.

Discussion of 9
Let X be compact metrizable and let Y be a continuous image of X. Then Y is compact. By point 7, Y has a countable network. By point 5, Y is metrizable.

Discussion of 10
A space is perfectly normal if it is normal and that every closed subset is a G_\delta-set. Let X be a space with a countable network. The normality of X comes from the fact that it is regular and Lindelof. Note that X is also hereditarily Lindelof. In a hereditarily Lindelof and regular space, every open subspace is an F_\sigma-set (thus every closed set is a G_\delta-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 Y=T \cup S where
T=\lbrace{(x,y) \in \mathbb{R}^2:y>0}\rbrace and
S=\lbrace{(x,y) \in \mathbb{R}^2:y=0}\rbrace.

Points in T have the usual plane open neighborhoods. A basic open set at p \in S is of the form B_c(p) where B_c(p) consists of p and all points q \in Y having distance <c from p and lying underneath either one of the two straight lines in Y which emanate from p and have slopes +c and -c, respectively.

It is clear that Y is a Hausdorff and regular space. The relative “Bow-tie” topologies on T and S coincide with the usual topology on T and S, respectively. Thus the union of the usual countable bases on T and S would be a countable network for Y. On the other hand, Y is separable but cannot have a countable base (hence not metrizable).

Reference
[Michael]
Michael, E., \aleph_0-spaces, J. Math. Mech. 15, 983-1002.

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3 thoughts on “Spaces With Countable Network

  1. Looks interesting Dan. You are quite adept with the Latex. I didn’t know that wordpress had that capability.

    I don’t know anything about Network topology. I shall read with interest. Many of the terms and theorems are familiar from Point-Set topology.

    It might be helpful to list some books or papers for further reading.

    • Your suggestion about books and papers for further reading is a great one. I will keep this in mind for future posts. In terms of text books, [Engelking] and [Willard] are handy source of basic information for me.

      [Engelking] Engelking, R., General Topology, Revised and completed edition, 1989, Heldermann Verlag Berlin.
      [Willard] Willard, S. General Topology, 1970, Addision-Wesley Publishing Company.

  2. A direct proof that nw(Sorgenfrey) is c. Suppose N be a network for the Sorgenfrey line. Pick, for each x, N_x in N such that x in N_x and N_x subset [x, x+1).
    Then if x != y, say x = c.
    We obviously have a base (so network) of size c, so nw(X) = w(X) = c.

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