How do you define sample space? My staff room is having a debate about the construction of sample spaces.

When you toss a coin twice, do you consider the sample space to be
  $$\{H,H\}, \{H,T\}, \{T,T\}$$ or $$\{H,H\}, \{H,T\}, \{T,H\},\{T,T\}$$

In my humble opinion, I feel there is no single correct answer at the moment because we do not have enough information. My feeling is that a sample space can only be established here if the order is relevant to the question at hand. In the absence of this information, either one could be the sample space.
However, I would like the community's thoughts on this. Is there in fact a single correct answer? Is there a mathematical reason for it being that answer?
 A: It depends what the sample space is for. 
If you are tossing 2 coins and just counting the number of heads/tails that happened then your first sample space would be correct. I.e you don't care about the order that the events occurred.
If you are tossing 2 coins and recording the 1st outcome and 2nd outcome separately then the second sample space would be correct. I.e. you do care about the order that the events occurred.
Alternatively, you could only be recording whether a head was flipped at all with either coin in which case the sample space becomes (with your notation):

{0}, {H}

So to summarize, a sample space is defined as the set of all possible measured outcome. So it's contents depend upon what you are measuring.
A: From my book: 

The set of all possible outcomes is the sample space corresponding to an
  experiment

The key word is an experiment. That is the sample corresponds to events that are possible for pertaining to experiment. I.e an example 

The number of jobs in a print queue of the mainframe computer may be
  modeled as

$$ \Omega = \{ 0 ,1, 2, ,3 \cdots\} $$

the set of all non-negative integers. However, in practice, there is
  likely upper limit $N$ on it.

$$ \Omega = \{ 0 ,1, 2, ,3, \cdots, N \} $$
