What does $D^1$ mean?

This might just be a stupid question but I saw that a torus can be represented as $S^1 \times D^1$ and I know that $S^1$ is a sphere of dimension $1$, so what is $D^1$?

edit: thanks for the answers

Solution 1:

The one dimensional disc, also known as a closed interval:

$D^1 \cong [0,1]$

Remember that in general, a manifold is called n-dimensional if every neighborhood requires n-coordinates to be parameterized, So $S^1$ is a circle and $D^2$ is a closed disc.

Solution 2:

In general $D^n= \{x \in \Bbb R^n\mid \|x\|_n \le 1\}$, the unit disk of $n$ dimensions (the norm being the usual Euclidean one with the square root of the sum of squares in general, and absolute value in one dimension). Its boundary (so with $\|x\|_n= 1$ instead), is called $S^{n-1}$ (it has one topological dimension less.) So for $n=1$ we get $[-1,1]$ (homeomorphic to any closed non-trivial bounded interval in $\Bbb R$) and $S^0 = \{-1,1\}$ a two point discrete space.

To visualise $S^1 \times D^1$, image a circle $S^1$) in the $xy$-plane and for each point of the circle a copy of $D^1 = [-1,1]$ vertically attached to it, going parallel to the $z$-axis. So we get $\{(x,y,x)\mid x^2+y^2=1; -1 \le z \le 1\}$ which is the "mantle" of a cylinder in $\Bbb R^3$ (no top or bottom disk). This is what topologists call a cylinder.