**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*,*x*_{0}) 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 *x*_{0} to *x* and the action of π(*X*, *x*_{0}) is by concatenation of paths. It is uniquely determined as a covering space.

Read more about this topic: Fundamental Group

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