Semisimple linear algebraic group Let $G$ be a linear reductive algebraic over an algebraically closed field of characteristic $0$. I know that there is a surjective map with finite Kernel $$G' \times T \to G$$ where $G'$ is semisimple and $T$ is a torus.
When I talk about semisimple/simply connected groups, I refer to their characterization based on their root system as in Springer book "Linear algebraic group".
I would like to prove that for every semisimple algebraic groups there is a finite covering $$G'' \to G'$$ such that $G''$ is  simply connected, but I really do not know how to do.
Also ,if it is possible, I would like to see a reference for the proof of the fact that over $\mathbb{C}$ a simply connected group is simply connected in the standard topological sense.
 A: Just to get this off the unanswered list.
For a proof that every semisimple group has a universal cover, at least in the split case, one can see [Mil, §18.d]. The general existence case can be handled by Galois descent.
Suppose now that $G$ is a semisimple group over $\mathbb{C}$. You want to show that the obvious map
$$\pi_1(G^\mathrm{an},e)\to \pi_1(G,e)$$
is an isomorphism. To do this we note that this map is induced by a map of Galois categories
$$\left\{\begin{matrix}\text{Central isogenies}\\ \text{over }G\end{matrix}\right\}\to \left\{\begin{matrix}\text{Finite topological covers}\\ \text{of }G^\mathrm{an}\end{matrix}\right\}$$
given by $G'\to G$ maps to $(G')^\mathrm{an}\to G^\mathrm{an}$. By abstract theory (e.g. see [Stacks, Tag0BN6]) it suffices to note that if $G'$ is connected then so is $(G')^\mathrm{an}$, but this is well-known (e.g. see [3, XII Proposition 2.4]).
From this, we see that it suffices to show that $\pi_1(G^\mathrm{an},e)$ and $\pi_1(G,e)$ have the same finite cardinality. But, to do this it suffices to show that the universal cover $\widetilde{G}$ of $G$ has simply connected analytification, since then
$$\pi_1(G,e)=\ker(\widetilde{G}\to G)(\mathbb{C})=\ker(\widetilde{G}^\mathrm{an}\to\widetilde{G})=\pi_1(G^\mathrm{an},e)$$
For this, see [Con, Proposition D.4.1] and note that the root data of semisimple gorup being simply connected is equivalent to the group itself being simply connected by the isogeny theorem (e.g. see [Mil, Theorem 23.9]).
[Con] Conrad, B., 2014. Reductive group schemes. Autour des schémas en groupes, 1, pp.93-444.
[Mil] Milne, J.S., 2017. Algebraic groups: The theory of group schemes of finite type over a field (Vol. 170). Cambridge University Press.
[Stacks] Various authors, 2020. Stacks project. https://stacks.math.columbia.edu/
