Can the sample space contain an impossible outcome? From my textbook:

We roll a standard six-sided die. Then the sample space is $\Omega=$
$\{1,2,3,4,5,6\}$. Each sample point $\omega$ is an integer between 1
and 6 . If the die is fair then each outcome is equally likely, in
other words $$ P\{1\}=P\{2\}=P\{3\}=P\{4\}=P\{5\}=P\{6\}=\frac{1}{6}.
$$ A possible event in this sample space is $$ A=\{\text { the outcome
is even }\}=\{2,4,6\} \text {. } $$ Then $$
P(A)=P\{2,4,6\}=P\{2\}+P\{4\}+P\{6\}=\frac{1}{2}. $$ The probability
measure $P$ contains our assumptions and beliefs about the phenomenon
that we are modeling.
If we scratch away the five from the original fair die and turn it
into a second two, the appropriate probability measure is $$
Q\{1\}=\frac{1}{6},\, Q\{2\}=\frac{2}{6},\, Q\{3\}=\frac{1}{6},\,
Q\{4\}=\frac{1}{6},\, Q\{5\}=0,\, Q\{6\}=\frac{1}{6} . $$ It is
perfectly valid to assign a probability of zero to a nonempty event,
as with $Q$ above.

Question: If the $5$ was scratched off the die, then wouldn't the sample space $\Omega$ change to $\{1,2,3,4,6\}$ since now $5$ could no longer be an outcome of the experiment?  And if so, then wouldn't it be true that there is no such event $\{5\}$ to even assign a probability to?
 A: As @user3716267 said in the comments, it's fine to have non-empty events with probability $0$, and it's pretty rare to remove them from the sample space.  In this case, removing it wouldn't really hurt anything, but doesn't give any benefits either.  In other cases, removing events with probability $0$ could definitely create problems.  For example, if $X$ is normally distributed, then $P(X=x) = 0$ for all $x \in \mathbb{R}$, but we definitely don't want to remove all the events $\{X=x\}$ or there would be nothing left!
A: 
Question: If the $5$ was scratched off the die, then wouldn't the sample space $\Omega$ change to $\{1,2,3,4,6\}$ since now $5$ could no longer be an outcome of the experiment?  And if so, then wouldn't it be true that there is no such event $\{5\}$ to even assign a probability to?

Yes; yes.
Since the sample space of a probability experiment is defined as its set of all possible outcomes, impossible outcomes like 5 and 7 are not elements of the sample space.
However, a sample space can contain zero-probability outcomes: such outcomes are possible, but happen almost never, and such a sample space has infinitely many outcomes. (One such outcome is hitting the exact centre of a dartboard.)
