Initial PR for Direct limit of finite-dimensional Euclidean spaces

[GeoffreySangston]

#1738

(1) It might be better to erase the argument for real TVS and find a reference. Currently I reference the "universal property of the final topology", which fits the normal pattern of universal properties, but maybe "characteristic property of the final topology" or some other name is more conventional. The property is stated on Wikipedia, so maybe that could be worth linking to.

(2) Note that $X$ is the countable coproduct over $\mathbb{R}$ as real topological vector spaces. I did not emphasize that in the property file though I mentioned it in the issue page. It might be a good idea to do so.

(3) I didn't want to overload on properties but these are two of the big ones.

[GeoffreySangston]

I forgot to sync before pulling. I'll fix tomorrow. Edit: I did below. I don't know if this can be done via the github vscode online editor, so I cloned the repo finally and used the terminal, which is much more familiar to me.

[yhx-12243]

Please merge main first, maybe

[GeoffreySangston]

@yhx-12243 Thanks for catching those. I forgot to double check on the actual website (just looked good in the github vscode editor preview feature).

[prabau]

Started with the README file. Looks very good, lots of good information. Some minor comments below.

[prabau]

We could also mention that $X$ is a closed subspace of $\mathbb R^\omega$.

[GeoffreySangston]

@prabau I was wondering how to treat quasi-component part. That is included in $X$ being a quasi-component; Willard writes, "the quasicomponent of [a point] is the intersection of all open-closed subsets which contain [it]" (a consequence of his definition). However, being a quasi-component is probably a property a lot of people would have to look up, so I wasn't sure if even mentioning "quasi-component" made sense.

We could also just change the following sentence to:

Hence $\mathbb{R}^\infty$ is a path component and a closed subset of {S107}.

[GeoffreySangston]

I was also thinking about how to fold the final topology part into the open sets condition, and I think that was a good change.

[prabau]

Will think some more about the quasi-component thing. In the meantime, some comment about direct limits.

A direct limit is unique up to unique isomorphism as they are characterized by some universal property. So there are multiple ways to "define" a direct limit, but they are all equivalent in some strong sense. Furthermore, as mentioned in https://en.wikipedia.org/wiki/Final_topology:
"The direct limit of any direct system of spaces and continuous maps is the set-theoretic direct limit together with the final topology determined by the canonical morphisms."
That's quite easy to check directly.

Also, in the category Set, if we have a increasing sequence of sets: $X_1\subseteq X_2\subseteq\dots$, the direct limit in Set of the direct system of the inclusions is just the union of the $X_n$. That is nearly immediate to check.
So one way to obtain the direct limit in Top is to take the union of the $X_n$ together with the final topology ...
In other words, it's back to the first paragraph of the README.

All this to say that we should not emphasize the particular description of the direct limit as a quotient of a direct sum. I suggest to drastically simplify the second paragraph as suggested below.

[prabau]

For the paragraph about the box product, it would be convenient to modify the name of S107 (countable box product of reals) by incorporating a notation for the space. Then we could slightly simplify things by referring to that notation.

The notation is not standardized, but here are some taken from mathse and mo:
$\square_{\alpha\in J}X_\alpha$
and for S107 in particular:
(1) $\square^\omega\mathbb R$
(2) $\square\mathbb R^\omega$
(3) $\square_{n=1}^\infty \mathbb{R}$
(4) $\square_{n\in\omega}\mathbb R$

Not sure between (1) and (2). What do you think?

[prabau]

Re: #1767 (comment) I agree that we don't need to mention quasi-components. We can mention that $X$ is a connected component and a path component, and a closed subset of the box product space.
(Anyone who clicks on the mathse link will still see something about quasi-components.)

[prabau]

P240 (CW complex): the proposed CW structure does not seem to be one.
In particular, one would need $\mathbb R^1\setminus\mathbb R^0$ to be a disjoint union of (open) 1-cells, which it is: it's made up of just two cells. But these cells are not properly "attached" to the 0-skeleton.
To get $\mathbb R^1$ itself as a CW complex one needs countably many 1-cells, and also countably many 0-cells.
And for the higher dimensions, one needs a whole "mesh" of n-cells (for all n) in infinitely many dimensions. I would guess.

Is there is a precise reference one could cite?

[GeoffreySangston]

@prabau Oh wow you're right. That's just terrible. Turned my brain off for that one. I'll have to look around today.

• I finally found a good reference: Fritsch–Piccinini. "Example 1 The Euclidean space $\mathbb{R}^m$ $m \in \mathbb{N}$, is a locally finite CW-complex of dimension $m$. This follows Theorem 2.2.2, which gives the fact for products of CW complexes. Maybe they have something about the colimit part of this but I'm not seeing it.
• This MSE thread describes the process, but it is fairly incomplete as a reference, so I'll keep looking.
• Wikipedia briefly describes a standard CW structure on $\mathbb{R}^n$ in the bullet beginning "The standard CW structure on the real numbers...".

I believe I looked through all of the references I mentioned in the old thread and most or all of the other references mentioned and didn't find it.

[yhx-12243]

A CW decomposition for S106 is 𝑋ₙ = {(𝑥₁, 𝑥₂, …, 𝑥ₘ, 0, …) | all coordinates are integers except for at most 𝑛 coordinates}.
It seems obvious intuitively, and may need more elaboration to make it rigorous.

[GeoffreySangston]

@yhx-12243 What about the n-cells? (It's loosely described in the post I linked.)

I think a formal way to do it is to give the CW structure on $\mathbb{R}$ which has the integers as the 0-cells and the open intervals between them as the 1-cells. I.e. $X_0 = \mathbb{Z}$ and $X_1 = X = \mathbb{R}$. Then there's a construction of the product of CW complexes; see Proposition A.6 of Hatcher. We then identify each with a subspace of $\mathbb{R}^\infty$ (and their cells with subspaces... informing the reader that the cells "line up") and take the union.

(Edit: Actually Fritsch–Piccinini might be better for the product part. See #1767 (comment).)

[GeoffreySangston]

Okay after looking pretty hard, I haven't found a perfect reference which spells it all out in one place. Unless someone is eager to write the corrected version, I will do it today using the argument I give above. Today is somewhat busy but I should have time at night to do it.

I could also write it as an answer to that old MSE thread if that's better.

[yhx-12243]

@yhx-12243 What about the n-cells? (It's loosely described in the post I linked.)

In fact it's easy to imagine (the mesh), but tedious to write down explicitly.

[GeoffreySangston]

@yhx-12243 Oh I misread your comment. I thought you were just saying take all coordinates integers. I think that's a good way to do it. Does our definition or Hatcher Proposition A.2 conveniently verify that though?

$X_n \backslash X_{n-1}$ equals all eventually $0$ sequences where exactly $n$ coordinates are non-integers.

[GeoffreySangston]

Another thought. I think explicitly writing the CW structure on the n-torus might be easier, and then pull that back to $\mathbb{R}^n$.

[prabau]

The description in terms of integers is a good one. Given an element $x\in X$, suppose it has non-integer coordinates $x_i$ for $i\in K$ for some index set $K\subseteq\mathbb N$ of size $k$, and its other coordinates are all integers. Then $x$ belongs to a (unique) open $k$-cell $e_k$, consisting of the points with non-integer coordinates exactly at the same coordinates.

A chararacteristic map is easy to describe if we identify the closed $k$-ball $D^k$ with $[0,1]^k$. Easy to map $[0,1]^k$ to the right thing ..., keeping the integer coordinates fixed (lazy to write it all). But it does not seem bad at all.

Will take a look at the refs you mention.

[GeoffreySangston]

Yea I agree formalizing @yhx-12243's solution is the right way to go.

[GeoffreySangston]

For the paragraph about the box product, it would be convenient to modify the name of S107 (countable box product of reals) by incorporating a notation for the space. Then we could slightly simplify things by referring to that notation.
Not sure between (1) and (2). What do you think?

In order of personal preference: (1) = (3) > (4) >> (2). @Moniker1998 probably has a preference/good idea on this since they've been doing a lot of box product stuff.

[prabau]

I was also thinking of asking @Moniker1998. In the mean time, we can go with (1), and change later if necessary.

[prabau]

For the CW complex mathse post, I think it would be preferable to have different people doing the asking and doing the answering. That will elicit more responses. Want me to ask the question?

[prabau]

There is already a (different) post that asks this exact question: https://math.stackexchange.com/questions/2429396 ("Is $\mathbb{R}^\infty$ a CW complex?") So it's a matter of adding an answer there.

And since the question is very brief and the poster has been gone for many years, it should be fine to expand the question to clarify what the space is. I can do that part if I get to it before you. (have to be out for a few hours)

[Moniker1998]

@prabau @GeoffreySangston no particular preference

[prabau]

@GeoffreySangston I have added details to the question https://math.stackexchange.com/questions/2429396.

And just noticed: the definition we have here uses superscripts $x=(x^1,x^2,\dots)$ instead of subscripts. Was that intentional?
Also, "eventually $0$ sequences": I guess that's clear, right?

[GeoffreySangston]

@prabau Sorry I got pretty lazy tonight and didn't work on this. Hopefully tomorrow will be better.

I did the superscripts probably without thinking about it. We can switch to subscripts if it's more conventional for pi-base.

[prabau]

Take all the time you want. Nothing urgent here.

[prabau]

Suggested change

	Equivalently, $\mathbb{R}^\infty \subset \mathbb{R}^\omega$ has the subspace topology, where
	$\mathbb{R}^\omega$ is given the box topology; this is shown in {{mathse:3961052}}. Moreover,
	it is shown in {{mathse:5012784}} that $\mathbb{R}^\infty$ is a quasi-component of the origin in
	$\mathbb{R}^\omega$. Hence $\mathbb{R}^\infty$ embeds into {S107}
	as a path component.
	Equivalently, $\mathbb{R}^\infty \subset \mathbb{R}^\omega$ has the subspace topology
	when $\mathbb R^\omega$ is given the box topology;
	that is, $\mathbb R^\infty$ is a subspace of {S107} (see {{mathse:3961052}}).
	Moreover, $\mathbb R^\infty$ is a connected component and a path component of $\square^\omega\mathbb R$
	(see {{mathse:5012784}}).

After all, it didn't seem necessary to mention that $X$ is closed in the box product. It's clear, and also a connected component is always a closed set.