# Confusion over wording of classification of covering spaces

I'm a little confused interpreting the statement of the classification of covering spaces in Hatcher's text.

Let $$X$$ be path connected, locally path connected, and semi locally simply connected. Then there is a bijection between the base point preserving isomorphism classes of path-connected covering spaces $$p:(\tilde{X},\tilde{x_0})\to (X,x_0)$$ and the set of subgroups of $$\pi_1(X,x_0)$$. If basepoints are ignored, the correspondence gives a bijection between the isomorphism classes of path connected covering spaces and conjugacy classes of subgroups of $$\pi_1(X,x_0)$$.

The way I'm reading "the base point preserving isomorphism classes of path-connected covering spaces" is that you fix a base point $$x_1$$ in the covering space $$p_1:(X_1,x_1)\to (X,x_0)$$. We identify $$Q(X_1,x_1)$$ as the collection of all covering spaces $$p_2:(X_2,x_2)\to (X,x_0)$$ where there is a homeomorphism $$f:X_1\to X_2$$ where $$p_1=p_2\circ f$$. Then for each subgroup of $$\pi_1(X,x_0)$$, there is a corresponding $$Q(X_1,x_1)$$. But when he talks about ignoring the base point, does he mean now that if we keeps $$X_1$$ the same, but change $$x_1$$ to $$x_1'$$, then $$Q(X_1,x_1)$$ and $$Q(X_1,x_1')$$ correspond to conjugate subgroups of $$\pi_1(X,x_0)$$? Is it not possible that $$(X_1,x_1')\in Q(X_1,x_1)$$?

To each path-connected covering space $$p:(\tilde{X},\tilde{x}_0)\to (X,x_0)$$ assign the subgroup $$G(p) = p_*(\pi_1(\tilde{X},\tilde{x}_0))$$ of $$\pi_1(X,x_0)$$. Then the assertion is
1. Each subgroup $$G \subset \pi_1(X,x_0)$$ has the form $$G = G(p)$$ for some path-connected covering space $$p$$.
2. $$p,p'$$ are basepoint-preserving isomorphic covering spaces if and only if $$G(p) = G(p')$$.
3. $$p,p'$$ are isomorphic covering spaces if and only if $$G(p),G(p')$$ are conjugate.
In 3. we only require that there exists a homeomorphism $$f : \tilde{X} \to \tilde{X}'$$ such that $$f \circ p = p'$$; we do not require $$f(\tilde{x}_0) = \tilde{x}'_0$$.
You consider a fixed covering space $$p_1:(X_1,x_1)\to (X,x_0)$$ and define $$Q(X_1,x_1)$$ as the collection of all covering spaces $$p_2:(X_2,x_2)\to (X,x_0)$$ where there is a basepoint-preserving homeomorphism $$f:X_1\to X_2$$ with $$p_1=p_2\circ f$$. If you drop the reqirement "basepoint-preserving", you get a larger class $$Q'(X_1,x_1)$$. If you replace $$x_1$$ by $$x'_1$$ in the same fiber, you trivially have $$(X_1,x'_1) \in Q'(X_1,x_1)$$, but in general not $$(X_1,x'_1) \in Q(X_1,x_1)$$. If $$p_1$$ is a normal covering space, then it is true for all $$x'_1$$. Thus it is possible. But if $$G(p_1)$$ is a non-normal subgroup of $$\pi_1(X,x_0)$$, you will find $$x'_1$$ such that $$(X_1,x'_1) \notin Q(X_1,x_1)$$.