The space of unoriented lines in the plane is diffeomorphic to the open Möbius band. To see why, let "L"("θ") denote the line through the origin at an angle "θ" to the positive x-axis. For each "L"("θ") there is the family "P"("θ") of all lines in the plane that are perpendicular to "L"("θ"). Topologically, the family "P"("θ") is just a line (because each line in "P"("θ") intersects the line "L"("θ") in just one point). In this way, as "θ" increases in the range , the line "L"("θ") represents a line's worth of distinct lines in the plane. But when "θ" reaches 180°, "L"(180°) is identical to "L"(0), and so the families "P"(0°) and "P"(180°) of perpendicular lines are also identical families. The line "L"(0°), however, has returned to itself as "L"(180°) "pointed in the opposite direction". Every line in the plane corresponds to exactly one line in some family "P"("θ"), for exactly one "θ", for , and "P"(180°) is identical to "P"(0°) but returns pointed in the opposite direction. This ensures that the space of all lines in the plane – the union of all the "L"("θ") for – is an open Möbius band.