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Due to the fact that it's taught early on in school, $\mathbb{R}$ is one of the first topological spaces studied. But what are some topological oddities that make $\mathbb{R}$ "weird" relative to other topological spaces?

Fomalhaut
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    "Weird" in the sense that it has some unexpected and undesired properties that make it harder to study than other spaces, or "weird" in the (improper, in my opinion) sense of "remarkable", i.e. of a space that is somewhat of a choking point of some lines of reasoning? – Sassatelli Giulio Oct 25 '22 at 06:49
  • @SassatelliGiulio the first sense – Fomalhaut Oct 25 '22 at 06:56
  • It's not compact (which is a nice property to have). – PhoemueX Oct 25 '22 at 07:04
  • Not sure whether this is what you are looking for: there are "weird" subsets of the real line (from a topological perspective), for instance Bernstein sets. For examples even more weird you could have a look into A.W. Miller's article in the Handbook of Set-theoretic Topology "Special subsets of the real line". But this might be hard to understand, it deals a lot with independence results, hence it requires some knowledge about forcing etc. – Ulli Oct 25 '22 at 07:24
  • There's Heine-Borel: the compact subsets are closed and bounded. Same for $\Bbb R^n$. – calc ll Oct 25 '22 at 08:18
  • @codeofsilence Boundedness is not a topological notion, though. That's metric. – Arthur Oct 25 '22 at 08:20
  • Oh, yeah I guess that's right. But compactness is. I thought it was a good answer because it seems to be something special about $\Bbb R$, not all metric spaces. @Arthur – calc ll Oct 25 '22 at 08:25
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    The weirdnesss of $\mathbb{R}$ is that we can construct the basis consisting of subsets homeomorphic to itself. It seems that its topology is a priori given in the basis level. I think it's very strange and weird. Also, it doesn't have minimum or maximum for every subset even though we can approximate its element as possible as we can. In addition, it's uncountable space, but it can be constructed by countably many bases. – ChoMedit Oct 25 '22 at 08:44
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    “$\mathbb{R}$ is connected, while everyone knows that the good topological spaces are profinite.” /s – Aphelli Oct 25 '22 at 09:04
  • @chomedit The statement on basis doesn't seem particularly relevant, since basis of a topology are very complicated objects (even for $\Bbb R$: there isn't just the one you talk about) and we don't have a well-developed theory of what the homeomorphism classes of their elements can be. Maxima and minima are not a topological property, and approximating elements is arguably the very purpose of topology. – Sassatelli Giulio Oct 25 '22 at 09:13
  • @SassatelliGiulio Thanks for pointing out. Yes, I also agree that the existence of maxima and the minima of subset of the space is not a topologcal statement. That is somewhat related to the ordering. I was confused that the topology always induces an ordering of the element, whose converse is partially true. The last point is, about the completeness, but well, that is also not a purely topological concept, I guess. It's already mentioned above. But anyway, I always thought that the existence of such basis for $\mathbb{R}^n$ seems weird, even though there seems no mathematical worth. – ChoMedit Oct 25 '22 at 09:19

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I would take the stance that $\Bbb R$ (with the standard topology) is not weird in any way, and anything that separates it from any other space is a "weirdness" about that other space.

$\Bbb R$ is the archetypal topological space. Every other topological space is compared to it. It is used to define higher dimensional Euclidean topologial spaces and everything that comes from that (homology and homotopy, paths and path connectedness, CW complexes, etc.). The number line (or segments of it) is the most basic concept in any part of math that deals with continuums. Any mathematician's intuition about topology was forged in relation to $\Bbb R$, either as an example or as a contrast.

I admit, some of this might be hyperbole, but I still stand by my first paragraph.

Arthur
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The term "weirdness" is very subjective. Is it weird that $\mathbb{R}$ and say $\mathbb{R}\backslash\mathbb{Q}$ are equinumerous? Or is it weird that $\mathbb{R}$ is homeomorphic to $(0,1)$ which is its own proper subset? It certainly is for some people.

However for most mathematicians $\mathbb{R}$ itself is not really weird. In fact topologically it behaves suprisingly well. Unlike its higher dimensional cousins $\mathbb{R}^n$.

The classical example is the exotic $\mathbb{R}^4$. We know that there is only one differential calculus on $\mathbb{R}^n$ for each $n$ except for $n=4$, where we have infinitely many essentially different calculi (is that plural for calculus?). While this is not a strictly topological property, I think it is worth mentioning.

A strictly topological example is: if $\mathbb{R}^n$ is homeomorphic to $X\times Y$, then what can we say about $X$ and $Y$? Well Kyung Whan Kwun shows in "Product of Euclidean Spaces Modulo an Arc" paper that if $n\geq 6$ then forget about $X$ being $\mathbb{R}^m$ while $Y$ being $\mathbb{R}^{n-m}$, they don't even have to be manifolds. For more information on $\mathbb{R}^n$ decomposition read this: Decomposition of a manifold.

Another interesting, and a surprising property of $\mathbb{R}$ is that we can cover higher dimensions with it. For example for any $n$ there is a surjective continuous map $\mathbb{R}\to\mathbb{R}^n$, glued from so called space filling curves. Intuitively, you would assume it is not possible to cover a plane with a line, but you can.

freakish
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