What is $\mathbb{C} \backslash \{0\}$?

Question

At first I thought is it just $\mathbb{C}$ without the point $(0,0) = 0+i0$ i.e. a punctured disk, puctured at the origin

But after reading What does $(\mathbb C\backslash\{0\})\times\mathbb R$ mean? I am not so confident that I am correct. Because in that answer someone remarked $\{0\}$ as being a "symbol"...

Can someone please explain the meaning of this notation, and also $\mathbb{R}^2\backslash \{0\}$

Answer 1

$\mathbb C\setminus\{0\}$ is exactly the punctured disk you mentioned.

In your question, they cartesian-multiplied it with $\mathbb R$ producing "the set of points of the form $(a + bi,c)$ where $i^2 = -1$; $a,b$ and $c$ are real numbers; and at least one of $a$ and $b$ is non-zero."

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$\mathbb R^2\setminus\{0\}$ is also what you would expect, i.e. $(a,b)$ where at least one of $a$ and $b$ is non-zero.

Answer 2

what is the reason for your doubt? $\mathbb{C}\backslash \{0\}$ is, in fact just the punctured disc.

And, actually, $0+i0=0$.