On which of the following spaces is every continuous (real-valued) function bounded? i) $X_1 = (0, 1)$; ii) $X_2 = [0,1]$; iii) $X_3 = [0, 1)$; iv) $X_4 =\{t \in [0, 1] : t \mbox{ irrational}\}$.
(i) is not true: example $f(x)=\frac 1x$ . (ii) I think this is true as the interval is closed and bounded. (iii) is not true. Example: $f(x)=\frac 1{1-x}$. (iv) I think this is true as this is a subset of (ii).
Am I right?