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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.)

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

5 Answers 5

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  • 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.

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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.

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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.

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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.

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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).

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