Difference between two defintions of Heine-Borel Theorem In class my notes gave the defintion of 
Theorem (Heine- Borel): If {U λ: λ∈Ω} covers [0,1] then there is a finite set λ  1... λn such that [0,1] ⊂ U  λ  i    : {1≤i≤n}
which he states means 
"Every open cover of [0,1] has a finite subcover."
But my textbook gives the defintion:
A subset E of R k is compact if and only if it is closed and bounded.
Could someone point out why this two are the same , because it seems to me that they are saying different things: specifically the textbook definition gives us if and only if but the definition my professor gave us is only one directional. 
I know this is not the place to clarify something like this but I have been pondering over for days.. thank you! Pardon my English, Dutch is my first language.
 A: The first of these is not the Heine-Borel theorem - in fact this is typically given as the definition of compactness! Other definitions, such as "$A\subseteq X$ is compact if every sequence in $A$ has a subsequence convergent in $A$", are equivalent to the one given, that every open cover of $A$ has a finite subcover.
The Heine-Borel theorem is particular to $\mathbb{R}^n$, and is exactly the second statement. Every compact set is both closed and bounded, but the converse is in general not true, and a common counterexample can be found by examining subsets of certain function spaces. The fact that $\mathbb{R}^n$ satisfies the important property that compactness is equivalent to closedness and boundedness is what the Heine-Borel theorem states.
(However, I should note that I have seen textbooks that call the first statement the Heine-Borel theorem as well, but I believe the convention I have mentioned above is the one most prevalent in analysis; if someone refers to the Heine-Borel theorem, they are almost certainly referring to the second statement and not the first.)
