Added: In this answer I take the sample space to be the full "Bernoulli space" $\mathcal{B} = \{T,H\}^{\infty}$. As Qiaochu Yuan points out in a comment to my other answer, this is a valid way to go: we identify a finite sequence $T^n H$ with the subset of infinite sequences with initial segment $T^n H$, and with respect to the natural (product) measure on $\mathcal{B}$, this event has probability $2^{-n-1}$, as it should. This is a more complicated space than the discrete space appearing in my other answer, but it is arguably more natural: the other space was contrived to answer a very specific problem about Bernoulli trials, whereas this space is the space of all Bernoulli trials.
You could ask the same question about any element of your sample space: "Why should we bother to have $(H,T,T,H,T,H,H,H,T,\ldots)$ in it -- after all, it only occurs with probability zero."
In this case, every singleton set occurs with probability zero, but if you took them all out you would have no sample space!
Based on your other recent question, I think you should start learning about measures and countable additivity. This has been the mathematical underpinning of probability theory for almost $80$ years.
Note also that the Bernoulli space $\mathcal{B}$ has a beautiful structure -- it can be viewed as $(\mathbb{Z}/2\mathbb{Z})^{\infty}$ and thus endowed with both a group structure and a compatible topology, under which it becomes a compact, totally disconnected abelian topological group. You wouldn't just start pulling points out of a compact topological group, would you? That will ruin everything...