Yes, if a set $A$ is compact, it must be closed and bounded in Euclidean space (or more generally, in a finite-dimensional normed vector space).

Explanation:

1. Compactness in Euclidean Space: A set is compact in a Euclidean space if every open cover of the set has a finite subcover. In simpler terms, it is "small" enough to be covered by a finite number of open sets and has no "leakage" (i.e., it contains all its limit points).

2. Closed:

   • A compact set in Euclidean space must be closed because if it were not, there would exist limit points of the set that are not contained within the set, contradicting the definition of compactness.

3. Bounded:

   • A compact set must also be bounded because if it were not, you could construct an open cover using balls of increasing radius that could never be reduced to a finite subcover, contradicting compactness.

Important Note:

While this is true for Euclidean spaces, the relationship between compactness, closedness, and boundedness can differ in other topological or infinite-dimensional spaces. For instance:

• In general topological spaces, compact sets are not always bounded (the concept of boundedness might not even exist).
• In an infinite-dimensional normed space, the Heine-Borel property (compact implies closed and bounded) does not hold without additional conditions.

Would you like me to explore related examples or explain these concepts further? Here are five follow-up questions:

1. What is the Heine-Borel theorem, and how does it relate to compactness?
2. How does compactness differ in general topological spaces compared to Euclidean spaces?
3. Can you give an example of a set that is closed and bounded but not compact?
4. Why is compactness useful in analysis, particularly in continuous function theory?
5. How does the definition of compactness change in infinite-dimensional spaces?

Tip: Always remember, in finite-dimensional spaces, compactness, closedness, and boundedness are deeply intertwined, but this connection weakens in infinite-dimensional or more general settings.