Choose the correct option.

Question

Let $X=\{(x_i)_{i\ge 1}:x_i\in \{0,1\}$ for all $i\ge 1\}$ with the metric $d((x_i),(y_i))=\sum_{i\ge 1}|x_i-y_i|2^{-i}.$ Let $f:X\to [0,1]$ be the function defined by $f(x_i)_{i\ge 1}=\sum _{i\ge 1}x_i2^{-i}$.

Choose the correct statements from below :

$1)$$f$ is continuous

$2)$$f$ is onto

$3)$$f$ is one-one

$4)$$f$ is open

My attempt : option $1)$ is true by triangle inequality.

$2)$ is true as it is a binary representation

im confused about option 3, 4

any hints/solution

thanks u

Answer 1

$f(1,0,0,\cdots)=f(0,1,1,\cdots)$ so $f$ is not one-to one. $\{(x_i):x_1=0\}\equiv \{(x_i):x_1 \neq 1\}$ is open and its image is $[0,\frac 1 2]$ which is not open. Hence $f$ is not an open map.