Etimology may help. Compact, from the Latin Compactus, past participle of the verb Compingere: "to pack together closely and firmly". You may have an idea of how strong is the hypothesis of compactness if you look at what happens when even total boundedness is lacking.

Just consider e.g. the possibly most familiar infinite dimensional object, the separable Hilbert space H=l₂. Its unit closed ball is not compact. A continuous real valued function on it may be unbounded, or bounded without minimum and maximum value. A linear form on H may be everywhere discontinuous and locally unbounded. There is a continuous transformation of the unit closed ball with no fixed points. The ball itself retracts on its boundary. Infinitely many disjoint unit open balls may be packed within a ball of radius $1+ \sqrt{2}$. The linear group GL(H) is connected. There are injective linear continuous transformations of H that are not surjective....

Etymology may help. Compact, from the Latin Compactus, past participle of the verb Compingere: "to pack together closely and firmly". You may have an idea of how strong is the hypothesis of compactness if you look at what happens when even total boundedness is lacking.

Just consider e.g. the possibly most familiar infinite dimensional object, the separable Hilbert space H=l₂. Its unit closed ball is not compact. A continuous real valued function on it may be unbounded, or bounded without minimum and maximum value. A linear form on H may be everywhere discontinuous and locally unbounded. There is a continuous transformation of the unit closed ball with no fixed points. The ball itself retracts on its boundary. Infinitely many disjoint unit open balls may be packed within a ball of radius $1+ \sqrt{2}$. The linear group GL(H) is connected. There are injective linear continuous transformations of H that are not surjective....