Debate about Cardinality of Reals vs. Cardinality of $[0,2\pi)$ My teacher claimed that the cardinality of the points on the unit circle (i.e. $[0,2\pi))$ was strictly greater than the cardinality of $\mathbb{R}$. I suspect he was mistaking the necessity of the existence of a bijection with the existence of a continuous bijection (which obviously wouldn't exist here). Am I right or am I missing something? The whole crux of his argument rested on a vague geometric argument and intuitively I could see why he'd think that but there is nothing intuitive about cardinals from what I've seen so far. Someone please put this to rest so I can finally sleep.
 A: If that is indeed what your teacher said, they are absolutely wrong. $[0, 2\pi)$ and $\mathbb{R}$ have the same cardinality, as you correctly observe.
It is interesting to note that there is a continuous injection from $\mathbb{R}$ to the circle but not conversely. The idea of looking at a "continuous" version of cardinality is an interesting one, and one particular take on it (continuous preimages) leads to the Wadge hierarchy, which is fantastically important in descriptive set theory. But, this is a completely different notion than cardinality.
A: The cardinality of $(-\frac{\pi}{2},\frac{\pi}{2})$ equals the cardinality of $\mathbb{R}$ because $x \mapsto \tan x$ is a bijection. 
Moreover, every two bounded open intervals $(a, b)$ and $(c, d)$ have the same cardinality because $f: (a, b) \rightarrow (c, d)$ given by $x \mapsto c+(d-c)\frac{x-a}{b-a}$ is a bijection. So $\mathbb{R}$ and $(0, 2\pi)$ have the same cardinality and clearly adding the point $0$ does not change the cardinality of $(0, 2\pi)$
