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