[anseladdams]

> But there is nothing in the definition that forces us to do that.

That's true. You're right.

> If the things were as you state would you have any use for Omega and Omicron then? Wouldn't just Theta suffice?

Couldn't there be cases where we don't have a known tight asymptotic bound but do have an upper and/or lower bound? And although it's an abuse of notation, you do often see big-O used in place of Theta. From CLRS:

"In the literature, we sometimes find O-notation informally describing asymptotically tight bounds, that is, what we have defined using Theta-notation."