I am preparing for Master's admission interview. I found some previous year interview questions.
- Is there a continuous map from $[0,1]$ to $(0,1)$? Why/Why-not?
My attempt: $f(x) = \frac{1}{3}$. So, answer is yes.
- Is there a continuous map from $(0,1)$ to $[0,1]$?
My attempt: $f(x)= 0$ if $x\in (0, 1/4]$, $12x-3$ when $x\in [1/4, 1/3]$, and $1$ when $x\in [1/3,1)$.
- Construct a bijective map from $(0,1)$ to $[0,1]$.
My attempt: Let $A=\{\frac{1}{2},\frac{1}{3},...\}$,$B=\{1,\frac{1}{2},\frac{1}{3},...\}$. Define $f:A\rightarrow B$ such that $f(\frac{1}{n})=\frac{1}{n-1}$. Now define a function $g:(0,1) \rightarrow [0,1]$ such that $g(x)=x$ if $x$ is not in $A$ , otherwise $g(x)=f(x)$.
Then $g$ is a the bijection from $(0,1)$ to $(0,1]$.
Am I right?