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:
“Poets ... are the only people to whom love is not only a crucial, but an indispensable experience, which entitles them to mistake it for a universal one.”
—Hannah Arendt (1906–1975)
“Three forms I see on stretchers lying, brought out there untended
lying,
Over each the blanket spread, ample brownish woolen blanket,
Gray and heavy blanket, folding, covering all.”
—Walt Whitman (1819–1892)
“A set of ideas, a point of view, a frame of reference is in space only an intersection, the state of affairs at some given moment in the consciousness of one man or many men, but in time it has evolving form, virtually organic extension. In time ideas can be thought of as sprouting, growing, maturing, bringing forth seed and dying like plants.”
—John Dos Passos (1896–1970)