The topology of the Möbius strip

Here is a Möbius strip that you can rotate and zoom with your mouse.

This picture should help you to understand the definition of the Möbius strip. It is swept out by the blue line as the green circle rotates around the z-axis. While the circle makes a full rotation, the line makes only half a turn.

This picture shows that the Möbius strip is homotopy equivalent to the circle.