Quiz, week 6

Let A=(0,1)A = (0, 1). In each case, what is f(A)f(A)? Some of these questions make no sense – please identify them.
- $f(x) = x$
- $f(x, y) = (x, y)$
- $f(x, y) = (x², y²)$
- $f(x) = (\cos πx, \sin πx)$
- $f : \mathbb{R}_{++} → \mathbb{R}$ is the Bertrand competition example, namely $f(x) = \begin{cases} \tfrac{1}{x} & \text{if } x < 1, \\ \tfrac{1}{2} & \text{if } x = 1, \\ 0 & \text{if } x > 1. \end{cases}$
Let A=(0,1)A = (0, 1). What is f−1(A)f^{-1}(A) when
- $f(x) = x$
- $f(x, y) = x² + y²$
- $f(x, y) = x$
- $f$ is the Bertrand competition example (see above)
Find a disconnected metric space (X,d₂)(X, d₂) such that AA is both open and closed, when
- $A = [0, 1]$
- $A = (0, 1)$
- $A = [0, 1] × (0, 1)$