We take [ilmath]\mathbb{RP}^2[/ilmath] (represented here as the gluings [ilmath]A[/ilmath] and [ilmath]B[/ilmath]) and cut it (creating gluings [ilmath]C_1[/ilmath] and [ilmath]C_2[/ilmath]), thus diving [ilmath]A[/ilmath] into [ilmath]A_1[/ilmath], [ilmath]A_2[/ilmath] and [ilmath]A_3[/ilmath]. The central strip is called [ilmath]M[/ilmath]
[ilmath]M[/ilmath] is a Mobius strip inside [ilmath]\mathbb{RP}^2[/ilmath]