I recently read Kosinski's Differential Manifolds and came across $\pi_1(X) = 1$ a few times. I know what the fundamental group is, I know that if the fundamental group corresponds to a singleton set, it is trivial. In other words, $X$ is simply connected.
But I am not sure whether $\pi_1(X) = 1$ is equivalent to $\pi_1(X) = \{pt\}$ or $\pi_1(X) = 0$. I have not too much experience in algebraic topology to spot equivalent meanings of different notations.
Thanks for any help!
As the name itself says, the fundamental group is a group. The notation $\pi_1(X)=\{pt\}$ has no sense if you don't endow the group with a topology. On the other hand $\pi_1(X)=0$ means that the group is trivial, that is contains only the neutral element. Such an element, is denoted with $0$ in the additive notation. In the multiplicative notation is denoted with $1$. It is a purely conventional fact here.