Open Real Interval is Homeomorphic to Real Number Line/Proof 2

Theorem

Let $\R$ be the real number line with the Euclidean topology.

Let $I := \openint a b$ be a non-empty open real interval.

Then $I$ and $\R$ are homeomorphic.

Proof

Let $I := \openint {-\dfrac \pi 2} {\dfrac \pi 2}$ denote the open real interval from $-\dfrac \pi 2$ to $\dfrac \pi 2$.

Consider the real function $f: I \to \R$ defined as:

$\forall x \in I: \map f x = \tan x$

Then we have:

$\forall x \in \R: \map {f^{-1} } x = \arctan x$

From Homeomorphism Relation is Equivalence it follows that $I$ and $\R$ are homeomorphic.

Then by Open Real Intervals are Homeomorphic, $I$ is homeomorphic to every other open real interval.

$\blacksquare$

Sources

• 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 7$: Subspaces and Equivalence of Metric Spaces: Exercise $2$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): homeomorphism
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): homeomorphism