Countable chain condition in topology

Question

A topological space $X$ is said to have the countable chain condition (ccc) if every collection of open and disjoint subsets of $X$ is at most countable. This definition can be found in L. Steen, J. Seebach: Counterexamples in Topology. p.22.

I am looking for a book or an article that contain the well known properties about topological spaces:

• Separability of $X$ implies that $X$ is ccc.

• For a metrizable space $X$, the ccc property implies separability.

• A counterexample of a completely regular Hausdorff space that has the ccc property but is not separable. (The website topology.pi-base.org does offer an example of such an example, namely the Peng-Wu group. I could not find a counterexample in the Literature. Unfortunately the counterexample provided by Steen and Seebach is not completely regular.)

Answer 1

Take a look at the table in the back of Steen and Seebach's book. You will find that Example 103 contains a completely regular space that is ccc but not separable: $\mathbb{N}^\lambda$, where $\lambda$ is a cardinal number larger than $2^{\aleph_0}$. The Tychonoff cube $[0,1]^\lambda$ provides a compact Hausdorff example.

Answer 2

How about the book Chain Conditions in Topology by W. Comfort and S. Negrepontis, Cambridge University Press (1982)? There's a lot of stuff in here.

For the third bullet point you can try Bell's A Normal First Countable ccc Nonseparable Space, or Tall's The Countable Chain Condition Versus Separability.

Answer 3

• Spearable implies ccc is theorem T21 of pi-Base, which references Counterexamples. The book doesn't provide a proof, but the result is standard. (Take a collection of pairwise disjoint open sets, if the space is separable then each contains a distinct point from the countable dense set, thus the collection is countable.)

• This pi-Base search maps out a proof that metrizable and ccc implies separable. The meat of this is theorem T62, showing every weakly Lindelof metrizable space is second-countable, based upon the Handbook of Set-Theoretic Topology. See also https://math.stackexchange.com/questions/4743591/.

• Once https://github.com/pi-base/data/pull/483 is reviewed and merged, @KP's suggestion from Counterexamples will be discoverable in pi-Base as space S1103.

Answer 4

A very nice example of a completely regular ccc non-separable space is the Pixley-Roy hyperspace of the reals. While it's not metrizable (and it can't be, as you already know) it is as close as it gets to being metric. In fact, it's a Moore space.

Answer 5

Just to add a couple of chapters in books.

Lectures on Set Theoretic Topology
M. E. Rudin
Softcover ISBN: 978-0-8218-1673-8
Product Code: CBMS/23
(available on Internet archive for 1 hour borrowing
https://archive.org/details/lecturesonsetthe0000rudi/mode/1up?view=theater)

In particular about ccc see:

Chapter III. Souslin Trees and Martin's Axiom

Also
Handbook of set-theoretic topology
Kenneth Kunen, Jerry E. Vaughan, 1984

(I think one could find info about ccc in various chapters, but in particular about cardinal invariants.)

Contents:\ R. Hodel, Cardinal functions. I (pp. 1–61); I. Juhasz, Cardinal functions. II (pp. 63–109); Eric K. van Douwen, The integers and topology (pp. 111–167); Scott W. Williams, Box products (pp. 169–200); Arnold W. Miller, Special subsets of the real line (pp. 201–233); S. Todorcevic, Trees and linearly ordered sets (pp. 235–293); Judy Roitman, Basic S and L (pp. 295–326); U. Abraham and S. Todorcevic, Martin's axiom and first-countable S- and L-spaces (pp. 327–346); Dennis K. Burke, Covering properties (pp. 347–422); Gary Gruenhage, Generalized metric spaces (pp. 423–501); Jan van Mill, An introduction to βω (pp. 503–567); Jerry E. Vaughan, Countably compact and sequentially compact spaces (pp. 569–602); R. M. Stephenson, Jr., Initially κ-compact and related spaces (pp. 603–632); Peter Nyikos, The theory of nonmetrizable manifolds (pp. 633–684); Franklin D. Tall, Normality versus collectionwise normality (pp. 685–732); William G. Fleissner, The normal Moore space conjecture and large cardinals (pp. 733–760); Mary Ellen Rudin, Dowker spaces (pp. 761–780); Teodor C. Przymusinski, Products of normal spaces (pp. 781–826); William Weiss, Versions of Martin's axiom (pp. 827–886); Kenneth Kunen, Random and Cohen reals (pp. 887–911); James E. Baumgartner, Applications of the proper forcing axiom (pp. 913–959); R. J. Gardner and W. F. Pfeffer, Borel measures (pp. 961–1043); S. Negrepontis [ Stelios Negrepontis], Banach spaces and topology (pp. 1045–1142); W. W. Comfort, Topological groups (pp. 1143–1263).

Answer 6

To the first point: This is Corollary 2.3.18 in "Ryszard Engelking. General topology. 1989."

To the second point: The following is a self-contained and elementary proof of this fact. If you need a reference, you will find the result as Corollary 4.1.16 in "Ryszard Engelking. General topology. 1989."

We assume without loss that $X$ is uncountable; otherwise there is nothing to show. Aiming for a contradiction, assume that $X$ is not separable. Let $\mathrm{d}$ be a metric on $X$, fix $\delta > 0$ and set \begin{align*} \mathcal{P}_\delta := \{P \subset E : \forall x, y \in P \colon x \neq y \Rightarrow \mathrm{d}(x, y) \geq \delta \} \subset 2^X. \end{align*} Ordered by inclusion, the set $\mathcal{P}_\delta$ is a partially ordered set. Moreover, it is straightforward to check that every chain in $\mathcal{P}_\delta$ has a maximal element. Hence, by Zorn's lemma, $\mathcal{P}_\delta$ contains a maximal element $P_\delta$. Now consider the union \begin{align*} P := \bigcup_{n \in \mathbb{N}} P_{1/n} \end{align*} and notice that $P$ is dense in $X$. Indeed, for every $n \in \mathbb{N}$ and every point $y \in X$, if follows that there exists $x \in P_{1/n}$ such that $\mathrm{d}(x, y) < \tfrac{1}{n}$. Otherwise, we may add $y$ to $P_{1/n}$, ending up with a larger set in $\mathcal{P}_{1/n}$ than $P_{1/n}$. This contradicts the maximality of $P_{1/n}$.

Since we assume the non-separability of $X$ and $P$ is a countable union of sets, there must necessarily be an $n \in \mathbb{N}$ such that $P_{1/n}$ is uncountable. Then the collection of open balls with $B(x, \tfrac{1}{3n})$ with radius $\tfrac{1}{3n}$ centred at $x \in P_{1/(3n)}$ contains uncountably many open disjoint sets. This contradicts the countable chain condition.