I have just discovered that a Haudorff compactification of a discrete space can fail to be completely regular in a model for ZF. A helpful paper for it is "Continuing horrors of topology without choice" by Good and Tree (Topol. Appl. 63(1995)); however, I have not seen any papers in which the discovery on compactifications mentioned above is included. I would be grateful if you could tell me whether my discovery is new for all concerned.
With kind regards. Eliza Wajch
I am not aware of any paper with such result, but it is interesting that in your model compact Hausdorff spaces need not to be normal, this is also nice. I would like to now the deductive strenght of the statement "Compact Hausdorff spaces are normal", as well !!!
Thank you for your answer very much. Please, notice that compact Hausdorff spaces are normal in ZF, so they are normal in every model for ZF but some of them can fail to be completety regular in some models for ZF because the Urysohn's lemma can fail for them in a model. I am glad that a Hausdorff not completely regular compact space in a model for ZF can have a completely regular dense subspace. This does not seem to be included nor trivially deduced from the paper by Good and Tree "Continuing horrors of topology without choice".
... Oh, I see ! The problem is with Urysohn's Lemma... But I confess I always thought that some choice was needed for "Compact Hausdorff spaces are normal", do you have any reference for your claim that this holds in ZF ? The usual proof of this needs a lots of arbitrary choices (pick for every point of one of the closed sets an open neighbourhood such that etc..), so I really don't see how to do it in ZF. Thanks.
... Hum, I thought a little more and now I see that "Compact T2 is T4" in fact holds in ZF. We do not have to pick exactly one open neighbourhood for every point of the closed sed; let us take them all, and then compactness make things go down to finite open sets and everything is OK. So, as usual, we first prove that compact T2 are T3 (in ZF, with this argument of picking all possible open sets for every point) and then repeat the argument for compact T3 is T4. Very nice ! Thank you.
It is necessary to work without the axiom of choice to understand more and be closer to truth. For the same reason, just to understand more and be closer to perfecness, it is also necessary to see what happens in models for ZF+the axiom of choice. Many unprovable in ZF important theorems have been discovered during one's research concerning models for ZF in which the axiom of choice fails. I do not believe that it is possible in physics to prove that there is an ordinal number of von Neumann which is equipollent with the "set" of all, for example, protons in the unique universe in which there is everything what really is. One the other hand, in ZF, the axiom of choice is equivalent with the statement that every set is well orderable. How to order well some special "sets"?
Now, this is a little bit confusing, but let me state a fact from point free topology (note: via Isbell's duality, the "arrows" get reversed in point free topology):
(a) for any frame L, the set Idl[L] of all ideals of L is a compact frame and the join map on Idl[L] is a dense frame homomorphism from Idl[L] to L
Thus this is a compactififaction of L which need not be regular!
The process of this construction does not involve choice at all and is just done within ZF.
Please let me know if I went wrong.
If a compact space is not regular, it is not Hausdorff. It holds true in ZF that every Hausdorff compact space is normal, so regular. However, a Hausdorff compactification of some discrete space need not be completely regular in a model for ZF. It is unprovable in ZF that every Hausdorff compactification of every discrete space is completely regular. I hope to prepare an article about it and make it available for other mathematicians soon.
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