I have the closed sets $[0,\infty]$ and $[0,1]$.
Define $f(x)=10^{-x}$ from $[0, \infty]$ to $[0,1]$.
I have these two sets and I have to prove that they are equal. It was easy in case of open sets but I am not sure this function that I have defined will work for closed sets because both 0 and 1 are included in codomain. Please, can someone help?
The function doesn't work since $\infty$ is not a real number and therefore the term $10^{-\infty}$ is meaningless.
However, it is not a problem to manually define $f(\infty)=0$, since $\infty$ is in the interval $[0,\infty]$. Moreover, since $10^{-x}\neq 0$ for all $x$, the function will remain bijection.