I was thinking about the following problem:
Let $X$ denote the two point $\{0,1\}$ and write $X_{j}=\{0,1\}$ for every j=1,2,3,....Let $Y=\prod_{j=1}^{\infty}X_{j}.$ Then, which of the following is/are true?
- $Y$ is a countable set,
- Card $Y$=card$[0,1],$
- $\bigcup_{n=1}^{\infty}$ ($\prod_{j=1}^{n}X_{j})$ is uncountable,
- $Y$ is uncountable.
Please help.Thanks in advance for your time.