What are the elements of metric space?Is it always an interval are it can be set of integers etc. What are the elements of metric space? Is it always an interval or it can be set of integers, etc.?
 A: An metric space can, as mentioned above, be any non-empty set. Hence, the elements of a metric space can be almost anything you can imagine, as long as you can equip that set with a metric. A metric is intuitively a way to measure the distance between elements. Here are some examples:
Example 1: $\mathbb{R}^n$ with the Euclidean metric. The elements are in this case vectors $\mathbf{x} = (x_1, \ldots , x_n)$ and the metric is
$$d(\mathbf{x}, \mathbf{y}) = \lVert \mathbf{x} - \mathbf{y} \rVert = \bigg( \sum_{j=1}^n (x_1-y_1)^2 \bigg)^{1/2}.$$
Example 2: The discrete metric. Let $X$ be a non-empty set. We can make it a metric space by endowing it with the discrete metric, given by
$$d(x,y) = \begin{cases} 
0, \quad & x = y \\
1, \quad & x \neq y 
\end{cases}.$$
Example 3: The vector space of continuous real-valued functions on $[a, b]$. Consider the vector space $C([a, b])$ of continuous functions $f : [a, b] \to \mathbb{R}$. We can give it with the following metric:
$$d(f, g) = \bigg( \int_a^b f(x)g(x) \, dx \bigg)^{1/2}.$$
Notice that in these three examples, the elements of the metric spaces are different objects. Mind you that a set can often be equipped different metrics and still be a metric set. For example are the metrics I have used in example 1 and 2 not the only metrics compatible with $\mathbb{R}^n$ and $C([a, b])$.
A: A metric space is an arbitrary non-empty set $X$, together with a metric, i.e., a map $d\colon X\times X\to\Bbb R$ with the appropriate properties.
