Find a bijection of the empty set: [0,1]--> (0,1).?

Question

I am pretty sure it is the empty set. The small o thingy with a line through it. This is an elementary analysis problem!

Answer 1

the empty set is just empty, and it has only one bijection...

you probably saw the greek letter phi: φ .

Answer 2

It's not the empty set, it's just the Greek letter phi (φ). The empty set has a similar but not identical symbol, but using the empty set doesn't make sense in this context. Here φ is just the name of the function.

The easiest way to do this is to pick a countable subset {x_n: n ∈ Z+} of (0, 1). Then map 0 to x_1, 1 to x_2, and x_n to x_(n+2).

For instance, let x_n = 1/2^n for n ∈ Z+. We get
φ: [0, 1] --> (0, 1)
φ(x) =
{1/2, if x = 0
{1/4, if x = 1
{x/4, if x = 1/2^n for some n ∈ Z+
{x, otherwise

You should be able to prove that this is a bijection.