We define the fundamental group by a homotopy of loops in a space X. However, this means we are considering path connected spaces. What about spaces that are completely disconnected? Do we say they do not have a fundamental group?
I have been thinking about the integers, specifically, and have not been able to find a concrete result. What are other spaces that would not have a fundamental group?
I have just been introduced to the fundamental group, so I am also rather curious about how we would write the fundamental group of a space that is only locally path connected with at least 2 components.
Remember that the fundamental group isn't really constructed from a space $X$, it's constructed from a space $X$ together with a distinguished point $x_0 \in X$. If $X$ is totally disconnected then the only loops in $X$ based on $x_0$ will be trivial ones, so $\pi_1(X, x_0) = \{e\}$ (i.e., the trivial group).
More generally, the same thing shows that if $x$ is an isolated point of any space then the fundamental group based at $x$ will be trivial, but the same space might have other fundamental groups that are not trivial.
It's only for path-connected spaces that we can always speak of "the" fundamental group of $X$.