Every compact set is totally bounded, but can we say that every totally bounded set is compact?
I'm a beginner in metric space. My thinking is that a totally bounded set behaves like a finite set and in some sense it is small. So it is very much like a compact set.
Someone please help me to clear my doubts. Thanks.
No, that is not enough (but almost).
Consider the set $(0,1)$ in the metric space $\Bbb R$, it is totally bounded but not compact. However, it is well known that totally boundedness & completeness is equivalent to compactness. You can read more about that here.