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Given a covering space $p : C \to X$ and a map $f : Y \to X$ from a path-connected and locally path-connected space $Y$, a lift of $f$ is a map $\tilde{f} : Y \to C$ such that $p \circ \tilde{f} = f$. The lifting property states that such a lift exists if and only if $f_*(\pi_1(Y, y_0)) \subseteq p_*(\pi_1(C, c_0))$.

A deck transformation, or covering transformation, for a covering space $p : C \to X$ is a homeomorphism $d : C \to C$ such that $p \circ d = p$. The set of all deck transformations forms a group under composition, called the Deck transformation group or group of covering transformations.

The homotopy lifting property asserts that given a homotopy $H: Y \times [0,1] \to X$ and a map $\tilde{f}: Y \to C$ such that $p \circ \tilde{f} = H(-, 0)$, there exists a unique homotopy $\tilde{H}: Y \times [0,1] \to C$ starting with $\tilde{f}$ such that $p \circ \tilde{H} = H$.

The path lifting property states that for each path $f: [0,1] \to X$ in a topological space $X$ and a point $c$ in a covering space $C$ such that $p(c) = f(0)$, there exists a unique path $\tilde{f}: [0,1] \to C$ in $C$ starting at $c$ such that $p \circ \tilde{f} = f$.

The monodromy action of the fundamental group $\pi_1(X, x_0)$ on a fiber $p^{-1}(x_0)$ is a group action where each loop in the base space starting and ending at $x_0$ determines a permutation of the points in the fiber above $x_0$ via lifting and following paths.

The Deck transformation group (also known as the group of covering transformations) of a covering space $p : C \to X$ is the group consisting of all homeomorphisms of $C$ that make $p$ commute, i.e., $p = p \circ d$ for each deck transformation $d$. This group is isomorphic to the quotient of the fundamental groups $\pi_1(X, x_0)/p_*(\pi_1(C, c_0))$ when $C$ is path-connected.

A covering space $p : C \to X$ is a regular (or normal) covering if the group of deck transformations acts transitively on each fiber. Equivalently, the covering is regular if $p_*(\pi_1(C, c_0))$ is a normal subgroup of $\pi_1(X, p(c_0))$.

The fundamental group of a topological space $X$, denoted $\pi_1(X, x_0)$, is the group of loops based at a point $x_0$ up to homotopy, with the group operation given by the concatenation of loops.

A covering space of a topological space $X$ is a topological space $C$ together with a continuous surjective map $p : C \to X$ such that for every $x \in X$, there exists an open neighborhood $U$ of $x$ where $p^{-1}(U)$ is a disjoint union of open sets in $C$, each of which is homeomorphic to $U$ by $p$.

A universal cover of a space $X$ is a covering space $p : C \to X$ where $C$ is simply connected (i.e., its fundamental group is trivial). The universal cover is unique up to homeomorphism and covers any other covering space of $X$.

A space $X$ is simply connected if it is both path-connected and every loop in $X$ can be continuously shrunk to a point (i.e., it has trivial fundamental group). Simply connected spaces are often used as universal covers for other spaces.