What set does $\mathbb W$ denote? 
What set does $\mathbb W$ denote?  

I know this may horribly lack context, but I've seen multiple times on M.SE $\mathbb W$ used in some fairly elementary context I think.
 A: In an elementary context, $\Bbb W$ means the set of whole numbers. Some books have it as
$$\Bbb W=\{0,1,2,\ldots\}$$
while others have it as
$$\Bbb W=\{1,2,\ldots\}$$
Because of the ambiguity, I recommend that you avoid the use of $\Bbb W$. For the second meaning use $\Bbb Z^+$. There still is no perfect abbreviation for the first. Either meaning is also called the Natural numbers, although usually the Whole numbers are meant to be different from the Natural numbers. Again, we see the ambiguity: even if the Whole and Natural numbers are different, which is which? See the Wikipedia articles for a variety of notations for these sets, most of which are far from perfect.
I have occasionally seen the phrase "whole numbers" used for the Integers, which includes negative numbers such as $-1,-2,\ldots$ and is usually written $\Bbb Z$. But I have never seen the notation $\Bbb W$ used in that way.
As @RenatoFaraone points out, in an advanced context $W(x)$ probably means the Lambert W function. But I have never seen that written in the "blackboard bold" style that $\Bbb W$ uses.
A: Just to provide a more or less authoritative reference as to what $\mathbb{W}$ denotes, the following is from page 2 of the book A Transition to Abstract Mathematics by Randall Maddox:

As you can see, $\mathbb{W}$ denotes the set of whole numbers, but this notation is often avoided in favor of $\mathbb{N}$, and even $\mathbb{N}$ itself is often clarified at the beginning of a text or mentioned in context whether or not $\mathbb{N}$ includes $0$. 
Also, I should mention that this book never makes use of $\mathbb{W}$ after page 2. It seems to be almost universally avoided and for good reason. 
A: The natural numbers, $\mathbb N$, are sometimes called the whole numbers, $\mathbb W$. It's ambiguous, because $-1$ is also a whole number, since it has no fractional parts.
