This question has been asked before.
Let me clarify what I don't get about each of the responses.
The first response (Michael Hardy): To me, it seems like this answer is comparing apples to oranges(no offense to the answerer: I am sure I am wrong). Michael defines a new random variable (X1,X2,X3), which looks bijective to me. However, he goes on to say that X1 is not bijective. This makes little sense to me.
The second response (Qiaochu Yuan):
$X^{−1}$ is not the inverse. It is the inverse image.
I believe the answer's saying that the Real numbers is infinite but the event space could be defined to have finite size, thus bijectivity doesn't apply because the size of codomain is larger than the size of the domain. While this is true, I can always change the set getting mapped to from real numbers to the numbers I map to. Ok, I am cheating a bit. Can anyone elaborate on this?
The third response (M Turgeon):
No, it doesn't have to be bijective - take a constant function.
Ok, this one logically makes sense. Unfortunately, overly constructed contradictions sometimes fail to change my instinct, and this is one of those cases. An example of a use case for a constant random variable would be helpful.
Please note that I am not familiar with Groups/Algebra, Topology or Analysis/Measure Theory.
What I want: I would like answers that expound on the answers to the question I have linked. Addressing the concerns I have mentioned in the question would be nice, but isn't necessary in interest of the response being useful to the community.
When you have a random vector $(X_1,X_2,X_3)$, each coordinate $X_i$ is a random variable. Michael Hardy is saying that the random variable $X_1$ is not injective.
You are right; you can always change the codomain to make the random variable surjective. Consider the opposite problem; suppose that the probability space $\Omega$ is infinite, while the range of the random variable is finite. Something like, flip a coin over and over until the result is heads, and let $X=1$ if the number of flips is odd and $X=0$ otherwise. The sample space is $\{H,TH,TTH,TTTH,\dots\}$, but the range is $\{0,1\}$. This random variable is essentially not bijective.
Imagine an urn with $100$ white balls and $101$ black balls. Over and over, remove two random balls from the urn. If they are the same color, throw in a white ball. If they are different colors, throw in a black ball. Continue (for 200 draws total) until there is only one ball left.
You would agree that the color of the last ball is a random variable, yes? It is the result of a series of random decisions. However, you can prove that no matter what sequence of random draws you make, the last ball will always be black! This is a constant random variable which arises "in the real world."