What is the exact definition of compactness

Question

I am confused with the definition of compactness.

My prof taught us this:

a subset U of $\mathbb{R}^d$ is compact if every open covering of U has a finite subcovering.

However, when I google the definition of compactness, it seems like the definition is this:

a subset is compact if it is closed and bounded.

Are they the same thing? If so, how are they related?

Answer 1

What your prof. taught is a definition of compact that works for a more general topological space,

not just $\mathbb R^n$. What you found via Google is true in $\mathbb R^n$ by the Heine-Borel theorem.

Answer 2

The standard definition is the one your teacher gave you. However, a subset of $\mathbb R^d$ is compact if and only if it is closed and bounded, Since, in general, this is easier to check, it is very much used in order to determine whether a subset of $\mathbb R^d$ is compact or not.

More generally, on any metric space $X$ the following conditions are equivalent for a subset $U$ of $X$:

• $U$ is compact;
• $U$ is complete and totally bounded.

Since a subset $U$ of $\mathbb R^d$ is bounded if and only if it is totally bounded, and it is closed if and only if it is complete, this is indeed a generalization of the equivalence from my first paragraph.