Mysior plane

[Moniker1998]

No description provided.

[Moniker1998]

In literature Mysior plane refers to two different spaces, one of which is not Tychonoff.
Not sure if this is a problem

[Moniker1998]

I believe this space is not countably paracompact, but I don't know so I'll just leave that property for future PR
I don't know if it's strongly zero-dimensional, I'll also leave that be.

[Moniker1998]

I need someone to give me a source that any uncountable subset of $\mathbb{R}$ contains a two-sided condensation point.
I.e. a point $x\in A$ such that for any $y < x < z$ the sets $(y, x)\cap A$ and $(x, z)\cap A$ are uncountable.

This is so I can show that Mysior plane is not para-Lindelof.

@prabau do you have any ideas?

I know it's true because I wrote a proof based on proof of theorem 2 in here: https://dantopology.wordpress.com/2009/09/25/a-countable-spread-property-unique-to-the-real-line/

[yhx-12243]

This is a very easy theorem @Moniker1998 so that it is unworthy to have a name with:
By contradiction for every x ∈ A there exists (x - δₓ, x + δₓ) which intersects A countably, then take its countable subcover (by hereditarily Lindelöfness of ℝ), we know that A is at most countable.

I don't know why dantopology take such long to prove it.

UPD:

Let Lₙ = { x ∈ A | |(x - 1/n, x) ∩ A| ≤ ℵ₀ }, Rₙ = { x ∈ A | |(x, x + 1/n) ∩ A| ≤ ℵ₀ }, then A = ⋃ (Lₙ ∪ Rₙ).
Hence at least one of them is uncountable, W.L.O.G Lₙ is uncountable.
By hereditary Lindelöfness of ℝ, exists countable B ⊆ Lₙ such that ⋃_(x ∈ B) (x - 1/n, x) = ⋃_(x ∈ Lₙ) (x - 1/n, x), so C := Lₙ \ ⋃_(x ∈ Lₙ) (x - 1/n, x) is uncountable.
However, x ∈ C ⟹ [x, x + 1/n) ∩ Lₙ = {x}. So C must be countable, a contradiction.

Still much shorter than dantopology.

[Moniker1998]

@yhx-12243 no, that only shows that there are condensation points

[Moniker1998]

I've added the property but I still need to add a citation there

[yhx-12243]

However, I think both P63 and P105 should be moved into mathse, since it is TOO long. (Can combine to one post, like that “More properties about Mysior plane”, like this) @Moniker1998

And it is also convenient to cite the two-side condensation lemma, like #1423 (comment) (updated).

[Moniker1998]

@yhx-12243 you can post there if you feel that way. I abstain from the rights to my proofs so feel free to copy those, or parts of them if you wish

[yhx-12243]

I suggest to open a mathse thread to discuss with and you can self-answer them, for these verbose proof. Then replace this commit to a reference to mathse, as we did earlier always.

[prabau]

Self-answered questions sometimes don't get as much attention in mathse. If you want, one of us can ask a question and then wait for people to answer. If after a day, nobody has given a good answer and you have a better one, you can post it also.

Let us know if you want us to ask a question there, and what you would like us to ask exactly.
(disclaimer: I have not read the discussion above in detail)

[prabau]

Is the question to get a short self-contained proof of the two-sided condensation lemma in $\mathbb R$?

[prabau]

https://math.stackexchange.com/questions/412547/every-bounded-non-countable-subset-of-mathbbr-has-a-two-sided-accumulation/412625
and specifically the answer of Brian Scott.

[Moniker1998]

@prabau oh, thanks! This proof looks exactly like the one by @yhx-12243

[Moniker1998]

@yhx-12243 to be clear, I won't post or answer a question, is what I meant in particular

[Moniker1998]

I note that I should rephrase locally metrizable using https://topology.pi-base.org/spaces/S000133
Also add that this space is locally orderable using this
The $\aleph$ and $\sigma$-space stuff will be taken care of in a new juicy PR

[Moniker1998]

@felixpernegger sure. Do note that I've added a bit much here so it'll be pretty long.

[mathmaster13]

another stuck PR
where is this PR stuck at?

However, I think both P63 and P105 should be moved into mathse, since it is TOO long. (Can combine to one post, like that “More properties about Mysior plane”, like this) @Moniker1998

And it is also convenient to cite the two-side condensation lemma, like #1423 (comment) (updated).

Was this ever posted? Moniker said someone else would do the post if we wanted to.

[prabau]

I have some comments about this one. Was planning to get to it, but did not have the time yet.

[prabau]

There is another well-known space from Mysior: "A regular space which is not completely regular" (https://pubs.ams.org/PROC/1981-081-04/S0002-9939-1981-0601748-4) = Engelking Example 1.5.9, also constructed by starting with a subset of points in the plane. (Initially, just from the name and before reading the contents, I mistakenly guessed that the space in this PR was referring to that space.)

If later on we introduce that other space, what would be good distinguishing names for the two spaces? Based on that, would there be a better name for the space in this PR? ("Mysior plane" does not seem a common name for this.)

[Moniker1998]

@prabau

Let me be honest. I think the non-completely regular Mysior space that's so popular is a useless example. As far as I know the only purpose was "to create an easy example" of a space in-between separation properties, and I don't think we need that. There already are examples of that here. And the Mysior space in this PR is useful but no one knows about it, and it's a completely obscure example because it doesn't appear online, only in some of my posts on mathoverflow and MSE, afaik.

I don't really care, I'm not planning to add the non-Tychonoff example. If you do, then that's your problem.

[felixpernegger]

@Moniker1998 do you still care about this space? If yes i can review this pr now, considering how long its been open

[prabau]

I am planning on reviewing it too.

[prabau]

@felixpernegger you don't have to review it if you don't feel like it (as you expressed earlier) :-)

[prabau]

I have replaced the P130 trait with one for P23. That makes the derivation for the locally compact related properties (P23, P24, P130) simpler.

[prabau]

P198 (countable extent = false) can be replaced by the stronger P227 (discrete closed set of size continuum). This will also automatically derive that Toronto = false.

[Moniker1998]

P198 (countable extent = false) can be replaced by the stronger P227 (discrete closed set of size continuum). This will also automatically derive that Toronto = false.

sure, the PR was made before the property was introduced to pi-base (I'll respond to the other comment at different time)

[Moniker1998]

P105 (not para-Lindelof): I am not quite sure I follow. What you are trying to say is maybe the following?

take x ∈ C so for every y < x the set ( y , x ) ∩ C is uncountable. That implies that every nbhd U m ( x + 1 ) of x + 1 meets uncountably many U n ( t ) with t ∈ C .

Isn't all that is needed? (but need to use U m ( x + 1 ) and not U m ( x ) .)

I mean I'm not trying to say this, but something similar, and this might be easier, sure.

[Moniker1998]

@prabau all your suggestions applied I believe

[prabau]

P63: what does "totally bounded" mean here?

[Moniker1998]

@prabau totally bounded. Real numbers are a metric space.

[prabau]

There is currently a circular dependency. X is Tychonoff (P6) is derived from Cech complete, but Cech complete (P63) itself requires Tychonoff and does not show Tychonoff itself.

So we should add a paragraph at the top of P63 showing why. I'll add a possible suggestion below.

[Moniker1998]

@prabau that's not true. Tychonoff is from Hausdorff + zero-dimensional. Maybe in the system it looks to be derived from that

[prabau]

Yeah, but we don't have Hausdorff as an asserted trait.

What is your proposal then?

[Moniker1998]

@prabau nevermind

[prabau]

Getting to the end of this beautiful space :-) I need to read Cech complete in more detail and that will be it.

[Moniker1998]

Getting to the end of this beautiful space :-) I need to read Cech complete in more detail and that will be it.

To be fair I think this space is quite basic of an example. At least that's my impression after all this time.