▁curve ▁in ▁the ▁plane ▁having ▁a ▁tang ent ▁line ▁at ▁each ▁point ▁determ ines ▁a ▁varying ▁line : ▁the ▁tang ent ▁bundle ▁is ▁a ▁way ▁of ▁organ ising ▁these . ▁More ▁formally , ▁in ▁algebraic ▁topology ▁and ▁differential ▁topology ▁a ▁line ▁bundle ▁is ▁defined ▁as ▁a ▁vector ▁bundle ▁of ▁rank ▁ 1 . ▁ ▁Line ▁bund les ▁are ▁specified ▁by ▁choosing ▁a ▁one - dimensional ▁vector ▁space ▁for ▁each ▁point ▁of ▁the ▁space ▁in ▁a ▁continuous ▁manner . ▁ ▁In ▁topological ▁applications , ▁this ▁vector ▁space ▁is ▁usually ▁real ▁or ▁complex . ▁ ▁The ▁two ▁cases ▁display ▁fund ament ally ▁different ▁behavior ▁because ▁of ▁the ▁different ▁topological ▁properties ▁of ▁real ▁and ▁complex ▁vector ▁spaces : ▁If ▁the ▁origin ▁is ▁removed ▁from ▁the ▁real ▁line , ▁then ▁the ▁result ▁is ▁the ▁set ▁of ▁ 1 × 1 ▁invert ible ▁real ▁matrices , ▁which ▁is ▁hom ot opy - equ ivalent ▁to ▁a ▁discrete ▁two - point ▁space ▁by ▁contract ing ▁the ▁positive ▁and ▁negative ▁re als ▁each ▁to ▁a ▁point ; ▁whereas ▁removing ▁the ▁origin ▁from ▁the ▁complex ▁plane ▁yields ▁the ▁ 1 × 1 ▁invert ible ▁complex ▁matrices , ▁which ▁have ▁the ▁hom ot opy ▁type ▁of ▁a ▁circle . ▁ ▁From ▁the ▁perspective ▁of ▁hom ot opy ▁theory , ▁a ▁real ▁line ▁bundle ▁therefore ▁behav es ▁much ▁the ▁same ▁as ▁a ▁fi ber ▁bundle ▁with ▁a ▁two - point ▁fi ber , ▁that ▁is , ▁like ▁a ▁double ▁cover . ▁ ▁A ▁special ▁case ▁of ▁this ▁is ▁the ▁orient able ▁double ▁cover