Can you describe what is $S^1 \times [0,\infty)$? Can you describe what is $S^1 \times$ [0,$\infty)$? Where "$\times$" stands for the product topology and the two factors are with the euclidean topology. 
EDIT: What about $S^1 \times S^1$?
 A: One way to think of the cross product, is that at each point in the first factor, you are attaching a copy of the second space.
For example: $S^1 \times \{s\}$, where $s$ is a single point, is really just $S^1$, since at each point you are replacing it with a different point.
$S^1 \times \{s,t\}$ is a two copies of the circle. You can view this by taking a circle, and at each point, you are replacing it with two points, and looking at the full collection of these gives two circles.
Actually, $S^1 \times \{1, \dots, n\}$ is nothing but $n$ circles, and $S^1 \times \mathbb Z$ is a countable collection of circles. You can visualize them as stacked along some verticle axis, with a circle at each integer.
Going further, $S^1 \times \mathbb R$ is a circle, but whenever there was a point, you replace it with a line, so you get a circle of lines, or in other words, a cylinder.
$S^1 \times [a,\infty)$ is the same, but with a half open interval.
$S^1 \times S^1$ is a circle of circles, so at each point you attach a circle (for the sake of visualization, say you attach a circle with smaller radius), then you get a torus, with  the traditional donut visual.
