Uniform structures on ℝ

Stream: maths

[Junyan Xu]

Anyone knows how many uniform structures are there on the real line that are compatible with its topology? There are at least four but are there more?

Update: at least five, it could be a circle minus a point :)

[Kenny Lau]

Junyan Xu said:

Anyone knows how many uniform structures are there on the real line that are compatible with its topology? There are at least four but are there more?

I don't know which four you're thinking about, but does it work if I use the continuous surjection R -> R^2 and then pull the metric back?

[Junyan Xu]

I don't think that induces the standard topology on R

[Junyan Xu]

What I have in mind are R, (0,\infty), (-\infty, 0) and (0,1).

[Junyan Xu]

Junyan Xu said:

Anyone knows how many uniform structures are there on the real line that are compatible with its topology? There are at least four but are there more?

Update: at least five, it could be a circle minus a point :)

Oh, there are non-metrizable uniformities on the reals: https://mathoverflow.net/a/41531/3332

[Aaron Liu]

Junyan Xu said:

Anyone knows how many uniform structures are there on the real line that are compatible with its topology? There are at least four but are there more?

Update: at least five, it could be a circle minus a point :)

surely there are infinitely many

[Junyan Xu]

Can we count them? Have I exhausted the metrizable ones?

[Junyan Xu]

Gemini gave the answer $2^{2^{\aleph_0}}$ but it's possibly hallucinated: https://g.co/gemini/share/a57ad9d6dbf8

[Aaron Liu]

you can embed it into a closed square [0,1]×[0,1] as a sinusoid

[Aaron Liu]

this seems metrizable
image.png

[Junyan Xu]

Yeah thanks, [0,1]×[0,1] is definitely metrizable. This construction gives a couple more distinct metrizable uniform structures as can be distinguished by the embeddings into the completions (closures), but I struggle to get infinitely many. (The limiting segments/points at the two ends could overlap with or be contained in the other or be joined at an endpoint.)

[Kenny Lau]

Is this related to the Stone-Cech compactification?

[Junyan Xu]

I don't see a connection; I can now see there are (countably) infinitely many distinct metrizable uniform structures though, since the two limiting segments can also touch in infinitely many combinatorially distinct ways ...

[Junyan Xu]

Okay I can see uncountably many now. Divide the two segments into countably many subintervals: on the even intervals let the intervals overlap; on the odd intervals you can choose whether they touch like 8 or like 0. This would give $2^{\aleph_0}$ many distinct combinatorial configurations.
Since the segments can also e.g. overlap along a Cantor set, there are probably much more.

[Notification Bot]

A message was moved here from #Is there code for X? > If X and Y are homeomorphic and X is complete, then Y is by Junyan Xu.

[Notification Bot]

A message was moved here from #Is there code for X? > If X and Y are homeomorphic and X is complete, then Y is by Junyan Xu.

[Aaron Liu]

Kenny Lau said:

Is this related to the Stone-Cech compactification?

A stackexchange post I read once (https://math.stackexchange.com/a/3088495):

Since you already know about the Alexandroff one-point compactification, let me begin by saying that the Stone-Cech compactification is at the other extreme, adding as many points at infinity as possible. To see what that could mean, let's consider some other compactifications of $\mathbb R$, starting with the most familiar, the extended real line, obtained by adjoining the two points $+\infty$ and $-\infty$ at the two ends of the line. Compared with the Alexandroff compactification, we're now distinguishing two different ways of "going to infinity". Some sequences that converged to $\infty$ in the Alexandroff compactification fail to converge in the extended real line because part of the sequence goes to the left and part to the right (e.g., $(-1)^nn$).

This idea can be extended, to produce "bigger" compactifications. If you visualize $\mathbb R$ as embedded in the plane as the graph of the sine function and then take its closure in the extended plane $(\mathbb R\cup\{+\infty,-\infty\})^2$, you get a compactification with a whole line segment at $+\infty$ (and another at $-\infty$). Similarly, embedding $\mathbb R$ in $3$-dimensional space as a helix, by $x\mapsto (x,\cos x,\sin x)$, we get a compactification with circles at the ends. Proceeding analogously with all bounded continuous functions $\mathbb R\to\mathbb R$ (in place of $\cos$ and $\sin$), in a very high-dimensional space (in fact, $2^{\aleph_0}$ dimensions), you get one of the standard constructions of the Stone-Cech compactification of $\mathbb R$. Roughly speaking, it separates, into different points at infinity, all of the possible "ways to go to $\infty$" in $\mathbb R$.

[Junyan Xu]

There is a uniformity the completion with respect to which gives the Stone-Cech compactification: https://mathoverflow.net/a/41504/3332
It's not metrizable though: https://math.stackexchange.com/questions/3403714/stone-cech-compactification-of-completely-regular-noncompact-space-is-not-metriz

[Junyan Xu]

@maintainers please move this to #maths

[Kenny Lau]

Junyan Xu said:

There is a uniformity the completion with respect to which gives the Stone-Cech compactification: https://mathoverflow.net/a/41504/3332
It's not metrizable though: https://math.stackexchange.com/questions/3403714/stone-cech-compactification-of-completely-regular-noncompact-space-is-not-metriz

I just meant that the "other" uniformities would give rise to "different" points of the Stone-Cech compactification, as mentioned by the above quoted post from StackExchange

[Junyan Xu]

The universal property of Stone-Cech gives a surjective closed map (therefore a quotient map) to any compactification, so you might know all compactifications if you understand the Stone-Cech compactification (not sure how you determine when a quotient space is metrizable though). However there are also completions that are not compact and maybe they are not captured by the Stone-Cech.

[Notification Bot]

This topic was moved here from #Is there code for X? > Uniform structures on ℝ by Oliver Nash.