Misconception about closed string worldsheet definition I'm a little confused about the precise way to define the worldsheet $\Sigma$ of a closed string. Its parametrization must be of the form $X: \Sigma \longrightarrow \mathbb{R}^{1,D-1}$ and one of its characteristic properties is the periodicity in $\sigma$ coordinate:
$$X(\tau, \sigma) = X(\tau, \sigma +2π) \tag1.$$
The problem is that some books define $\Sigma$ as the set $\mathbb{R} \times [0,π]$ (Becker & Becker), or $\mathbb{R} \times [0,2π]$ (Polchinski) and most of the books claim that $\Sigma$ has the topology of a cylinder. So, apply to X a $\sigma$-value outside the domain $[0,2π]$, for example, makes no sense.
I know that we usually define the circle $S^1$ as the quotient $\mathbb{R}/\sim$, where $\forall \sigma_1,\sigma_2 \in \mathbb{R}, \sigma_1\sim \sigma_2 \iff \sigma_2 = \sigma_1 +2πn, \ n \in \mathbb{Z}$.
What also would make no sense is define
$$X: \mathbb{R} \times S^1 \longrightarrow \mathbb{R}^{1,D-1} \tag2,
$$
since the elements of $S^1$ are sets, not numbers.
So, how do I define correctly the worldsheet and parametrization of a closed string?
 A: All the things you claim make no sense do, in fact, make sense.
The following are equivalent:

*

*A function $f_r : \mathbb{R}\to X$ with $f(x) = f(x+2\pi)$ for all $x$.

*A function $f_i : [0,2\pi]\to X$ with $f(0) = f(2\pi)$

*A function $f_s : S^1 \to X$
We start with a function $f_r$ as in $1$. Then the restriction $f_i = f_r\vert_{[0,2\pi]}$ is a function as in 2.
If we think of an element of the circle $S^1$ as an angle $\phi\in[0,2\pi)$ (in your representation as a quotient of $\mathbb{R}$, just choose the smallest positive number in each equivalence class as its representant), then $f_s(\phi) = f_i(\phi)$ is a function as in 3.
Finally, given an $f_s$ and again choosing the angle parametrization of $S^1$, define a function $f_r$ on $\mathbb{R}$ by $f_r(x) = f_s(x \mod 2\pi)$. This is a function as in 1.
Therefore, it does not matter whether people say that the worldsheet parameter $\sigma$ has the property $f(\sigma) = f(\sigma + 2\pi)$ for all $f$, that it is valued in $[0,2\pi]$ with $f(0) = f(2\pi)$ for all $f$ or that it is valued in $S^1$ - all these things describe the same situation, namely that of the circle $S^1$ and hence the closed worldsheet of the free string as a cylinder.
Note that people aren't always careful with their language, so if stuff like "$\mathbb{R}\times[0,2\pi]$ is the cylinder" bothers you because it is technically wrong you have to learn to live with that and extend some amount of good faith towards the authors (in this case that they really meant to glue the ends of the interval together to form an $S^1$).
