Universal Covering Space
If X is a topological space that is path connected, locally path connected and locally simply connected, then it has a simply connected universal covering space on which the fundamental group π(X,x0) acts freely by deck transformations with quotient space X. This space can be constructed analogously to the fundamental group by taking pairs (x, γ), where x is a point in X and γ is a homotopy class of paths from x0 to x and the action of π(X, x0) is by concatenation of paths. It is uniquely determined as a covering space.
Read more about this topic: Fundamental Group
Famous quotes containing the words universal, covering and/or space:
“A universal and perpetual peace, it is to be feared, is in the catalogue of events which will never exist but in the imaginations of visionary philosophers, or in the breasts of benevolent enthusiasts.”
—James Madison (1751–1836)
“We have good reason to believe that memories of early childhood do not persist in consciousness because of the absence or fragmentary character of language covering this period. Words serve as fixatives for mental images. . . . Even at the end of the second year of life when word tags exist for a number of objects in the child’s life, these words are discrete and do not yet bind together the parts of an experience or organize them in a way that can produce a coherent memory.”
—Selma H. Fraiberg (20th century)
“Why not a space flower? Why do we always expect metal ships?”
—W.D. Richter (b. 1945)