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) :-)

[Moniker1998]

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

@felixpernegger I mean, after all this time my interest has diminished slightly, but I do care. I'll check what Patrick suggested above after I actually finish checking with your PR.

[prabau]

Question after reading the explanation that X is Tychonoff in https://math.stackexchange.com/questions/4718866.
Seems to me each $U_n(x)$ is clopen in $X$. It's open and its complement is also open.
Am I missing something?

If that's the case, for a continuous function separating a closed set $F$ and a point $(x,0)\notin F$, one can just take the characteristic function of some $U_n(x)$.

[prabau]

P22 (pseudocompact): the justification relies on a metaproperty: "This property is hereditary with respect to clopen sets."
It's rather obvious, but still we could add it to P22.

P22 is not hereditary wrt to open sets. Also it is not hered. wrt to closed sets (example: S8 (particular point topology on an infinite set): is P22 but has an infinite closed discrete subspace).

ADDED LATER: Can you first rebase this PR on main? That will help out to avoid merge issues and will bring other useful meta-properties into the PR. If you have difficulties doing it, let me know and I can try to do it.

[prabau]

As mentioned in #1423 (comment), it will be good to rebase (or merge main) for this PR. This will give access to the meta-properties for the locally compact properties.

[Moniker1998]

Question after reading the explanation that X is Tychonoff in https://math.stackexchange.com/questions/4718866. Seems to me each U n ( x ) is clopen in X . It's open and its complement is also open. Am I missing something?

If that's the case, for a continuous function separating a closed set F and a point ( x , 0 ) ∉ F , one can just take the characteristic function of some U n ( x ) .

@prabau it's a post I've made 3 years ago, and I might have just went with an explanation that was easy for me at the time. Clopen sets weren't a focus.

Cozero complemented needs to be modified since it doesn't require Tychonoff anymore anyway, and $T_1$ or $T_0$ needs to be added.

[Moniker1998]

@prabau I did rebase it. Somehow it deleted whole S211 though

[Moniker1998]

@prabau I'd like you to suggest me some solution to this

[prabau]

I'll take a look.

[prabau]

It seems the first commit of this PR added the Mysior space with S211 (in Aug 2025). And the second commit changed it to S215 (also Aug 2025). In the mean time, space S211 was added to main in Dec 2025.
And then when doing the rebase to main, git takes the main version and replays on top of it each of the commits for this PR. So the first thing that does is move the S211 files to S215 and then keep appling more commits after that. The result is that S211 has disappeared. And then the latest force-pushed to this PR seems to have lost the original history.

@Moniker1998 I assume you did that rebase on a local branch. Would you have a copy of the branch before the rebase by any chance?

I am not sure how to resolve this, but I'll keep looking.

@yhx-12243 would you have an idea?

[prabau]

@Moniker1998 In the mean time, regarding the post https://math.stackexchange.com/questions/4718866/mysior-plane-is-not-realcompact, to show the $X_+$ is realcompact, wouldn't the following simpler argument also work ?

For the case $L=\mathbb{R}\times \{0\}\in\mathcal{F}$: the restriction of the real z-ultrafilter $\mathcal F$ to $L$ is a real z-ultrafilter on $L$. And $L$ has the discrete topology. So all we need to show is that a discrete space of cardinality continuum is realcompact. pi-base already knows that, but for the purpose of a simple proof, it can be shown directly.
Take the closed interval $Y=[0,1]$ with the discrete topology. Given a real z-ultrafilter $\mathcal F$ on $Y$, split $Y$ as the union of two closed intervals of length 1/2. One of these two intervals is in $\mathcal F$. So take that one and repeat the splitting, countably many times. The countable intersection of the appropriate intervals is a singleton, which belongs to $\mathcal F$. So the z-ultrafilter is fixed.

That seems a pretty simple proof. Does that seem right to you?

(But if that argument works, one could repeat the same argument for the whole space $X$ and again deduce that $X$ is realcompact, which it it not. So there must be something wrong somewhere.)

[Moniker1998]

@Moniker1998 I assume you did that rebase on a local branch. Would you have a copy of the branch before the rebase by any chance?

I am not sure how to resolve this, but I'll keep looking.

@prabau I don't know how to check that, I am using github desktop. Also an easy way to resolve it would be to add everything from S211 again, I guess.

[prabau]

Also an easy way to resolve it would be to add everything from S211 again, I guess.

We should not touch S211 in this PR. But one can rework this PR to add everything for S215 in one single commit. I tried it on a local branch on my machine. One loses the history of the various commits here, but that does not seem too important. I could do it if you want. What do you think?

[Moniker1998]

@prabau sure, no problem

[Moniker1998]

@Moniker1998 In the mean time, regarding the post https://math.stackexchange.com/questions/4718866/mysior-plane-is-not-realcompact, to show the X + is realcompact, wouldn't the following simpler argument also work ?

For the case L = R × { 0 } ∈ F : the restriction of the real z-ultrafilter F to L is a real z-ultrafilter on L . And L has the discrete topology. So all we need to show is that a discrete space of cardinality continuum is realcompact. pi-base already knows that, but for the purpose of a simple proof, it can be shown directly. Take the closed interval Y = [ 0 , 1 ] with the discrete topology. Given a real z-ultrafilter F on Y , split Y as the union of two closed intervals of length 1/2. One of these two intervals is in F . So take that one and repeat the splitting, countably many times. The countable intersection of the appropriate intervals is a singleton, which belongs to F . So the z-ultrafilter is fixed.

That seems a pretty simple proof. Does that seem right to you?

(But if that argument works, one could repeat the same argument for the whole space X and again deduce that X is realcompact, which it it not. So there must be something wrong somewhere.)

@prabau why would the restriction to $L$ be a real $z$-ultrafilter?

[prabau]

I removed the files from S211 that were accidentally copied into the S215 directory due to the rebase (and were breaking the build) and put all the S215 in a single commit. Will continue looking at it tomorrow. But at least S211 in now untangled from S215.

[prabau]

why would the restriction to L be a real z -ultrafilter?

The assumption is that L is a member of $\mathcal F$. So the restriction $\mathcal G$ of $\mathcal F$ to L is a z-filter. But maybe it's not a z-ultrafilter on L ? (Maybe that's where things fail: a zero set in L is not necessarily a zero set in X ?

So there could be zero sets in L that meet every zero set in L, but are not zero sets in X, so they would not belong to $\mathcal G$.

Summary: the intersection of two zero sets is a zero set. But a zero set within a zero set need not be a zero set ?

[Moniker1998]

@prabau well, yes, correct. A zero-set of a zero-set need not be a zero-set. There's a lot of those little conditions for subsets that regulate this and other things.

[Moniker1998]

@prabau by the way, github desktop now says "You have 28 local commits ..." and would like to push to github. Is there anything I can do about this, because I feel like it's trying to reset the progress done here by your commit.

[Moniker1998]

@prabau perhaps the best example is Jones lemma. If you interpret it right, it's a statement about zero-sets extending from a discrete closed set of size continuum.

So find yourself a discrete zero-set of size continuum in a separable space. You'll see one of the subsets cannot extend to a zero-set (and so is not a zero-set of the whole space).

[prabau]

@prabau by the way, github desktop now says "You have 28 local commits ..." and would like to push to github. Is there anything I can do about this, because I feel like it's trying to reset the progress done here by your commit.

Yeah, I have seen the same thing sometimes. The way I fix it is to delete my local branch and then recreate it (git checkout) from the main repository. So things are back in sync.

Or better, you can make a backup of your local branch, for example by changing the name of your local branch. Then you don't even have to delete it first and you have a record of the old commits. And then get a copy of the desired branch from main.

[Moniker1998]

@prabau alright then I think everything from the technical side has been done. All suggestions put forward have been applied. Anything else?

[prabau]

I have not reviewed the whole thing. Still need to look at the more complicated traits.