Bacle said:What is your definition ofS being precompact? The one I knew is that the closure of S is compact.
micromass said:I think the OP means totally bounded. I've seen this terminology used before...
Anyway, you are working in [itex]\mathbb{C}[/itex]. So try to prove that the closure of a totally bounded set is (totally) bounded. Then use Heine-Borel.
micromass said:I think the OP means totally bounded. I've seen this terminology used before...
Anyway, you are working in [itex]\mathbb{C}[/itex]. So try to prove that the closure of a totally bounded set is (totally) bounded. Then use Heine-Borel.
Compactness is a mathematical property of a set that indicates how much the set "fits" inside a larger space. In simpler terms, it measures how tightly the points in a set are clustered together.
"Iff" is a mathematical shorthand for "if and only if". In this context, it means that compactness and precompactness are equivalent, or they are both true at the same time.
Precompactness is a weaker version of compactness that is often used in metric spaces. It means that the set can be covered by a finite number of smaller sets, each of which has a finite diameter.
Compactness is closely related to the concept of limits. In a compact set, any sequence of points will have a limit point, or a point that the sequence "approaches" as the number of points increases. In a non-compact set, this may not be the case.
A few examples of compact sets are a closed interval on the real number line, a sphere in three dimensions, and a closed and bounded subset of a metric space. Non-compact sets include an open interval on the real number line, an infinite plane in two dimensions, and an unbounded subset of a metric space.