Add Q^ω, as a better distinguishment of Erdős space

[yhx-12243]

This is a continuation of #1141/#1142.

• The arguments of LOTS may tedious, it is possible to open an MSE post.
• Also Add the argument that $\mathbb R^\omega$ is not Menger.

[yhx-12243]

I'm not sure that we added LOTS currently because it may be not so obvious, or if we can make a mathse post.
The reason I keep metrizable is similar because it is much simpler than the argument of LOTS.

(Another offtopic: $\mathbb Q^\omega$ and $\mathbb Q \times \left( \mathbb R \setminus \mathbb Q \right)$ are not homeomorphic but pi-base-indistinguishable)

[StevenClontz]

Making sure I understand https://topology.pi-base.org/spaces/S000146/properties/P000133 - Theorem 2.7 of the referenced paper shows that S146 embeds as a dense subspace of the irrationals, and a dense subspace of a LOTS (like the irrationals) is a LOTS. Is that right?

[StevenClontz]

Let's defer the proof for LOTS for after a discussion on MSE.

[yhx-12243]

Let's defer the proof for LOTS for after a discussion on MSE.

Agree. I have not found any post that "dense subspace of a LOTS is again a LOTS" and I think it's right (it is not hard to prove), I worry that whether I make something wrong. 🤐

[prabau]

The two aliases for S146 don't seem to serve much purpose. Unless we can justify their existence here, they should be removed.

[prabau]

Since this PR used twice the fact that closed subspaces of Menger spaces are Menger, it would be good to have this as a meta-property. But it can be done later.

[prabau]

Actually, re-reading what is there to justify P66, what is written there makes little sense. The second sentence starts with "It follows from". That seems to mean the "It" refers to the previous sentence, which is not what you mean at all.

Let me rephrase the whole thing for one of them, and you can do the other one.

[prabau]

In the description for the new space, shouldn't we mention it's a subspace of $\mathbb R^\omega$ (S30)?

[prabau]

refs: section for S146 (and one other place): When referencing a paper from the literature or a book, we usually add the name of the author/authors in parentheses after the title, for the name: field.

See for example https://topology.pi-base.org/theorems/T000277/references

[prabau]

S146 - P53: The English again makes no sense. Please, fix the grammar.

[prabau]

S146 - P27: Bad grammar. Rephrase.

[prabau]

P133: I agree that the LOTS is too much and should be a post on mathse.

[prabau]

Sorry, but the grammar for P27, P53 still does not make sense. Do you see why?

[prabau]

Finally! Thank you :)

[prabau]

Apart from removing the trait for LOTS, these were all my comments.

[yhx-12243]

Let's defer the proof for LOTS for after a discussion on MSE.

Agree. I have not found any post that "dense subspace of a LOTS is again a LOTS" and I think it's right (it is not hard to prove), I worry that whether I make something wrong. 🤐

In fact the proposition is that "A (topological) dense subspace of a (order) dense LOTS is a LOTS", where two "dense"s has different meanings.

Proof: Let $X$ be a dense LOTS, $Y$ be a dense subset of $X$. Let $a \in X \setminus Y$, we will prove that $(a, \to)$ is open in order of $Y$.

Consider $\bigcup_{b \in Y, a < b} (b, \to) \subseteq (a, \to)$, for any $y \in Y$ such that $a < y$, it follows from order density of $X$ that $(a, y)$ is a nonempty open set in $X$, which must intersects $Y$ (by topological density of $Y$) with elements in $Y$ (e.g., $b$), then $y \in (b, \to)$.

I'll put it on mathse at some time.

[prabau]

Approved, but @StevenClontz needs approval too to unblock.

[StevenClontz]

Marking approved per @prabau review