I am studying compactness, and I need to be confirmed if I have categorized the definition of compactness correctly. I would appreciate if you take a look and see if I am correct in recognizing the definitions?
is this the DEFENITION of compact metric space $M$? $M$ is said to be compact, if $M$ is complete and totally bounded.
is this the DEFENITION of compact subset $A$ in a metric space $M$? $A$ is said to be compact, if every open cover of $A$ has a finite subcover.
is this the DEFENITION of compact subset $A$ in Real numbers? $A$ is said to be compact, if every sequence from $A$ has a subsequence converging to a point $a\in A$ .
are these correct?