Let C be a Cantor set, show that 1) every point of C is an accumulation point of C,2) every point of R\C not a

Let C be a Cantor set, show that 1) every point of C is an accumulation point of C,2) every point of R\C not a


Let C be a Cantor set, show that :


1) every point of C is an accumulation point of C


2) every point of R\C not an accumulation point of C

Let C be a Cantor set, show that 1) every point of C is an accumulation point of C,2) every point of R\C not a
For 2), a Cantor set is formed by removing a sequence of open sets from R, and so it is closed (can be written as an intersection of closed sets). Thus R\C is open, and so every point of R\C has a neighborhood contained in R\C, and so is in particular is not an accumulation point of C.





1) is more complicated, and I think I'll leave it to you. There is probably more than one way to do it, but I think I would argue by contradiction. Assume there is an isolated point in C and find a contradiction.

crab apple