Trigger switch using basic components
Yes. What you are describing sounds like a flip-flop. These are fundamental building blocks of digital circuits, and are available in great quantity at very low cost as ICs. It will be more expensive, and performance will be worse, to build them from discrete components. However, it can be done:



Pulling inputs E1 or E2 low will set or reset the outputs A1 and A2. You can see that if one transistor is on, it will turn the other off, so A1 is always the complement of A2.