* Inspect: "To look at something carefully and officially, especially to check that it is satisfactory." The "spec" part emphasizes the act of looking closely and critically.
* Suspect: "To believe that someone has committed a crime or done something wrong, but without having definite proof." Here, "spec" suggests the act of looking at someone with suspicion, or seeing them as potentially guilty.
* Spectator: "A person who watches an event, especially a sports event." "Spec" highlights the act of observing or watching an event, similar to the idea of looking or seeing.
In essence, "spec" in these words all relate to the act of looking or seeing, but with different connotations depending on the context.