What is the point of a lift in topology? I've just covered 'lifts' in topology and also homotopy lifting to a covering map but I'm struggling to understand the intuition behind lifts and essential the 'point' of them.
Could someone please explain this intuitively? Thanks!

There are many answers to this question, all pertaining to the fact that lifts are extremely important in homotopy theory. For instance, automorphisms of covering spaces can be used to calculate fundamental groups, and the homotopy lifting property is used to define fibrations. I don't want to bombard you with examples, so I'll leave it at this; but if you read ahead in your textbook you'll probably find liftings all over the place.
