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=== Properties === Let <math>X</math> be a path-connected space and <math>p:E \rightarrow X</math> be a connected covering. Since a deck transformation <math>d:E \rightarrow E</math> is [[Bijection|bijective]], it permutes the elements of a fiber <math>p^{-1}(x)</math> with <math>x \in X</math> and is uniquely determined by where it sends a single point. In particular, only the identity map fixes a point in the fiber.{{r|Hatcher|p=70}} Because of this property every deck transformation defines a [[group action]] on <math>E</math>, i.e. let <math>U \subset X</math> be an open neighborhood of a <math>x \in X</math> and <math>\tilde U \subset E</math> an open neighborhood of an <math>e \in p^{-1}(x)</math>, then <math>\operatorname{Deck}(p) \times E \rightarrow E: (d,\tilde U)\mapsto d(\tilde U)</math> is a [[group action]].
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