Can Someone Explain the Set $\{ ∈ ℝ^ ∣ d(x,0) = 1\}$? I'm trying to understand this set. I understand most of the individual parts such as $∈$ means "an element of", $ℝ$ is all real numbers, and $d(x,0) = 1$ is the distance from $x$ to $0$, as in absolute value.
I don't understand what $x$ means in this situation though. Is it considered a variable? Is it multiple values? 
I also don't understand what the actual elements of the set are. Is the value of $x$ and element in the set? Also, are there any other numbers with an absolute value of $1$ other than $1$ and $-1$? If not, are those the only possible values for $x$? 
In this situation, $N$ is a number of dimensions, but I don't understand how that relates to real numbers, nor what real numbers to any power would be.
Thanks.
 A: $x$ is just a name. Translated into English, the set
$$\{x \in \Bbb R^n | d(x,0) = 1 \}$$
would read "all elements of the set $\Bbb R^n$, such that the distance between that element and $0$ is $1$." (Presumably $0$ here denotes the "origin," so to speak, if you were considering graphing in $\Bbb R^n$.) This is essentially the $n$-dimensional unit sphere - for example, in $\Bbb R^2$, this set denotes all points on the circle of radius $1$.
The name $x$ is just a shorthand, it doesn't mean anything other than giving elements of $\Bbb R^n$ some sort of name in the setbuilder notation above.
$\Bbb R^n$, to answer your other question, is the $n$-dimensional analogoue of the real number line. For example, $n=2$ gives you the $xy$-plane; $n=3$ gives you typical $3D$ Cartesian space; and so on.
If you're at all familiar with the notion of Cartesian product, $\Bbb R^n$ can also be expressed as
$$\mathbb{R}^n = \underbrace{\mathbb{R} \times \mathbb{R} \times \mathbb{R} \times ... \times \mathbb{R} \times \mathbb{R}}_{\text{n times}}$$
and thus that allows us to express elements of $\Bbb R^n$ by $n$-tuples. For example, for $n=2$, we would have our typical $(x,y)$ ordered pairs from the $xy$-plane. Generally then, for the $n$-dimensional case, you could express elements of $\Bbb R^n$ by ordered $n$-tuples $(x_1, x_2, x_3, ..., x_n)$, where $x_k$ is a real number for all $k$.
A: The notation $d(x,0)=1$ seems to indicate that this is a problem in a general metric space. This is not necessarily the absolute value, as there can be more abstract metrics. This set is the unit ball in $\mathbb{R}^N$ with respect to this metric. 
