Prove that an interval without endpoints is homeomorphic to the real number line \(\mathbb{R}\). Thus, boundedness is not a topological invariant. Take \(X=\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\) and \(Y=\mathbb{R}\), and consider the map \(f: X \rightarrow Y\) given by f(x)= tan x.