Weak fundamental group

The fundamental group has a reasonable meaning for path connected spaces. The weak fundamental group generalizes the classical notion and applies to connected spaces. If the space is path connected, then the weak fundamental group is isomorphic to the classical one. By using the new notion, we prove the existence theorem for the universal covering space. Also, we give an example of calculating the weak fundamental group for a space that is not path connected. The weak fundamental group can be defined in a purely algebraic way. In our preprint https://arxiv.org/abs/1904.13130, we define a fudamental group for a class of $C^*$-algebras. Under some conditions, the weak fundamental group of a space $X$ is naturally isomorphic to the fundamental group of the $C^*$-algebra $C_0(X)$.