Need Help about Lebesgue Covering Dimension

duck_duckone

guys i'm currently working on a paper that contains of topological dimension. as i dashed around the internet i still haven't figured out the way to define the dimensions. and if it's not much of a burden for you guys, could you explain to me in simple words. thanks guys

1.Cantor's proof that there is a one-to-one correspondence between R1 and R2. would you please explain?
2. Peano's construction of a continuous map from R1 onto R2?
3. how do you define that a koch curve is R1 while a square is R2 by the coverings? and could you please give me another example than the koch curve and square

thanks a lot guys.