What are some topological oddities of $\mathbb{R}$? 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?
 A: 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.
A: 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.
