Galois correspondence for covering spaces

Let \(X\) be path connected, locally path connected, and semilocally simply connected, with basepoint \(x_0\). Then:

• Isomorphism classes of based connected covering spaces \[ (p:E\to X,e_0),\qquad p(e_0)=x_0, \] correspond bijectively to subgroups \[ H\le\pi_1(X,x_0), \qquad H=p_*\pi_1(E,e_0). \]
• If basepoints upstairs are forgotten, connected covering spaces correspond to conjugacy classes of subgroups of \(\pi_1(X,x_0)\).
• The covering is regular (normal/Galois) exactly when \(H\trianglelefteq\pi_1(X,x_0)\). In general its deck group is \[ N(H)/H, \] and for a regular cover this is \(\pi_1(X,x_0)/H\).
• The universal cover corresponds to the trivial subgroup. If \(\widetilde X\to X\) is universal, every connected cover is obtained, up to equivalence, as \[ \widetilde X/H\to X \] for a subgroup \(H\le\operatorname{Deck}(\widetilde X/X)\cong\pi_1(X,x_0)\).