Are $\pi(X,x)=0$ and $\pi$$(X,x)=${1} denoted the same thing, the trivial group? Both notation come from a same book: A first course in algebraic topology by Kosniowski.
Note that $\pi(X,x)$ is the set of equivalence classes of closed paths based at $x\in X$. That is the fundamental group of $X$ with base point $x$.
If they are the same, why the author use different notation? If they are not the same, what are the differences of them?
Here it is an example of true statement in the book:
If $X$ is a finite topological space with the discrete topology, then $\pi(X,x)=0$. (Page 125)
A topological space $X$ is simply connected if it is path connected and $\pi$$(X,x)=${1} for some (and hence) any $x\in X$.(Page 130)
If $G$ is any group, then the statements $G=0$ and $G=\{1\}$ both mean that $G$ is the trivial group.