Sample space in probability: Subsets of the sample space in "different dimension"? Recently in my class I have encountered the question as to whether the following statement is considered correct:
Define $\Omega = \{ 0,1 \}^\mathbb{N} $,
are subsets of  $\{0,1\}^n$ for $ n = 1,2,3,...$ also subsets of $\Omega$?
It seems strange if the answer is yes because if I consider 
$\{0,1\}^2$ = $\{(0,0), (0,1), (1,0),(1,1)\}$, and 
$\{0,1\}^1$= $\{(0),(1)\}$ then intuitively it feels very strange to compare $\{(0)\}$ with $\{(0,1)\}$ or $\{(1,0)\}$ because they seem to have different 'dimensions', if the use of such term is even appropriate here.
However the question I have been given seems to suggest that the above is correct.
Could somebody enlighten me on this question? Thanks! 
Furthermore, if the above holds true, would elements in the sigma algebra of, say $\{(0,1)^1\}$ be elements in the sigma algebra of $\{(0,1)^2\}$ as well? Thanks!
Edit: Subject to the response that I have just received to the above question to be 'no'. Would it appear that my assignment question is making a contradiction? Is it perhaps defining this fact simply for purpose of this question? 
Source of wonder.image

Let us consider one simple example. We have an infinite sequence of coin  flips. For this case $\Omega=\{0,1\}^{\mathbb N}$. Are the words "first two flips will both be tails" describe an event in $\Omega$? This looks as the same question as above: whether $\{(0,0)\}$ is a subset of $\Omega$ or not. But really these are two different questions. 
The words "first two flips will both be tails" describe the set of all 0-1 sequences which begin with two zeros: $$(0,0)\times\{0,1\}^{\mathbb N}.$$ This set is the subset of $\Omega$. But the set $\{(0,0)\}$ is not.
I suppose that your question should sounds smth like: are the results of (infinite series of) coin flips that say something about first $n$ tosses only, the subsets of $\Omega$? This question will be answered as "Yes". 
