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:

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 $X$ , $nw(X)=w(X)$ . In particular, any compact space with a countable network is metrizable.
As a corollary to 5, $w(X) \leq \vert X \vert$ for any compact $X$ .
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).
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 $T_3$ (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 Responses to Spaces With Countable Network

Robert Meyer

November 14, 2009 at 7:48 pm

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.

- Dan Ma
  
  November 14, 2009 at 8:46 pm
  
  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.
  
Henno

March 30, 2010 at 11:28 am

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.

Spaces With Countable Network

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