# Countable chain condition in topology

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.)

• Problem N on page 60 of J. L. Kelley, General Topology, has a hint. Nov 27, 2023 at 19:16
• See the Ryszard Engelking's classic monograph on General Topology. Dec 6, 2023 at 4:50

• 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.

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.

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.

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.

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)