I was reading Hatcher's Algebraic topology book, there is a statement about the fundamental group on different path connected component as follows:
Since $\pi_{1}\left(X, x_{0}\right)$ involves only the path-component of $X$ containing $x_{0}$, it is clear that we can hope to find a relation between $\pi_{1}\left(X, x_{0}\right)$ and $\pi_{1}\left(X, x_{1}\right)$ for two basepoints $x_{0}$ and $x_{1}$ only if $x_{0}$ and $x_{1}$ lie in the same path-component of $X$.
This statement is reasonable,I try to make it a theorem and prove it,is it possible to prove this statment?For example for homology group on different component the direct sum of them isomorphic to the holomogy group of the whole space.
Assuming you want to prove that if there is a path $\gamma$ between points $x_0$ and $x_1$ in $X$, then $\pi_1(X,x_0) \cong \pi_1(X,x_1)$, you can do the following:
Let $\gamma : [0,1] \to X$ be a path with $\gamma(0)=x_0$, $\gamma(1)=x_1$. Define $$ \gamma_* : \pi_1(X,x_0) \to \pi_1(X,x_1), \quad [w] \mapsto [\gamma * w * \overline{\gamma}], $$ where $\overline{\gamma} : [0,1] \to X$, with $\overline{\gamma}(t) := \gamma(1-t)$, and $$ (\gamma * w * \overline{\gamma})(t) = \begin{cases} \gamma(3t) & 0 \leq t \leq \frac13\\ w(3t-1) & \frac13 \leq t \leq \frac23\\ \gamma(3(1-t)) & \frac23 \leq t \leq 1 \end{cases}. $$ The map $\gamma_*$ is well-defined, because if $w \sim w'$ are homotopic via a homotopy $H$, then we get $\gamma * w * \overline{\gamma} \sim \gamma * w' * \overline{\gamma}$, via the Homotopy $H' : [0,1] \times [0,1] \to [0,1]$, with $$ H'(t,s) = \begin{cases} \gamma(3t) & 0 \leq t \leq \frac13\\ H(3t-1,s) & \frac13 \leq t \leq \frac23\\ \gamma(3(1-t)) & \frac23 \leq t \leq 1 \end{cases}. $$ Next, we see that $\gamma_*$ defines a group homomorphism, since $$ \gamma * (w * w') * \overline{\gamma} \sim \gamma * (w * (\overline{\gamma}*\gamma) * w') * \overline{\gamma} \sim (\gamma * w * \overline{\gamma}) * (\gamma * w' * \overline{\gamma}), $$ where we used that $\overline{\gamma}*\gamma$ is homotopic to the constant path $c_{x_1}$ at $x_1$.
Lastly, we show that $\gamma_*$ is an isomorphism. We know that $\overline{\gamma}$ induces a map $\overline{\gamma}_* : \pi_1(X,x_1) \to \pi_1(X,x_0)$. We want to show that $\gamma_* \circ \overline{\gamma}_* = \text{id}_{\pi_1(X,x_1)}$. Take some $[w] \in \pi_1(X,x_1)$. Then, we get that $$ (\gamma_* \circ \overline{\gamma}_*)([w]) = [ \gamma*\overline{\gamma}*w*\gamma*\overline{\gamma} ] = [c_{x_0}*w*c_{x_0}] = [w]. $$ Similarly, one shows that $\overline{\gamma}_* \circ \gamma = \text{id}_{\pi_1(X,x_0)}$.
Therefore, $\gamma_*$ is an isomorphism, so $\pi_1(X,x_0) \cong \pi_1(X,x_1)$. Hope this was helpful!