Open Real Intervals are Homeomorphic

Theorem
Consider the real numbers $\R$ as a metric space under the Euclidean metric.

Let $I_1 := \left({a, b}\right)$ and $I_2 := \left({c, d}\right)$ be open real intervals.

Then $I_1$ and $I_2$ are homeomorphic.

Proof
Consider the real function $f: I_1 \to I_2$ defined as:
 * $\forall x \in I_1: f \left({x}\right) = c + \dfrac {\left({d - c}\right) \left({x - a}\right)} {b - a}$

Then after some algebra:
 * $\forall x \in I_2: f^{-1} \left({x}\right) = a + \dfrac {\left({b - a}\right) \left({x - c}\right)} {d - c}$

By the Combination Theorem for Continuous Functions, both $f$ and $f^{-1}$ are continuous on the open real intervals on which they are defined.

Hence the result by definition of homeomorphism.