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(Rotman p282, intro to algebraic topology) Let $(\tilde{X},p)$ be a covering space of $X$, $x_0, x_1 \in X$ and let $Y_0$ , $Y_1$ be the fibers over $x_0,x_1$ respectively. Then $|Y_0|= |Y_1|$.

Proof:

$$\require{AMScd} \begin{CD} \pi_1(\tilde{X}, \tilde{x}_0) @>\Sigma>> \pi_1(\tilde{X}_1, \tilde{x}_1) \\ @Vp_* V V @VVp_* V\\ \pi_1(X,x_0) @>> \sigma > \pi_1(X,x_1) \end{CD} $$

Let $\tilde{\lambda}$ be path from $\tilde{x}_0$ to $\tilde{x}_1$, and $\lambda = p \tilde{\lambda}$. $\Sigma$ sends $[\tilde{f}] \mapsto[ \tilde{\lambda}^{-1} \cdot \tilde{f} \cdot \tilde{\lambda}^{-1}]$, and $\sigma$ sends $[f] \mapsto [\lambda^{-1} \cdot f \cdot \lambda]$. Since $p_*$ is an injection, it follows that $\Sigma$ induces a bijection between cosets $$ [ \pi_1(X,x_0): p_*\pi_1(\tilde{X},\tilde{x}_0)]=[ \pi_1(X,x_1): p_*\pi_1(\tilde{X},\tilde{x}_1)]$$

I don't understand the bolden part of the proof: how is the bijection induced? My thoughts:

Let the left cosets of $\pi_1(X,x_0)$ wrt $p_*\pi_1(\tilde{X},\tilde{x}_0)$ denoted by $[a]_0$ and the other by $[a]_1$. The map $$[a]_0 \mapsto [\sigma(a)]_1$$ is well-defined: If $[a]_0= [a \cdot f ] _0$, $f \in p_*\pi_1(\tilde{X}, \tilde{x}_0)$, then $[\sigma(a \cdot f)]_1 = [ \sigma(a) \cdot \sigma(f)]_1 = [\sigma(a)]_1$ from commutativity.

Similarly, we have the inverse map $[a]_1 \mapsto [\sigma^{-1}(a)]_0$.

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