Advanced calculus was the first course where I ran into definitions that I struggled to understand in a serious way. What does it mean for the interval [0, 1] t

Advanced calculus was the first course where I ran into definitions that I struggled to understand in a serious way. What does it mean for the interval [0, 1] to be "compact"? Well, it means exactly what the definition says (e.g., every open cover has a finite subcover), but what did that "mean"? This was the first course where the definitions were pushed beyond the boundary of my intuition. Seems like a way of thinking that would be useful for computer engineering, or damn near anything else.