Real-Valued Mapping is Continuous if Inverse Images of Unbounded Open Intervals are Open/Proof 2

Theorem

Let $T = \struct {S, \tau}$ be a topological space.

Let the real number line $\R$ be considered as a topology under the usual (Euclidean) topology.

Let $f: T \to \R$ be a real-valued function on $T$.

Then:

$f$ is continuous

if and only if

for all $a \in \R$: $f^{-1} \openint \gets a$ and $f^{-1} \openint a \to$ are open in $T$.

Proof

From Sub-Basis for Real Number Line:

$\set {\openint \gets a, \openint b \to: a, b \in \R}$ is a sub-basis for $\R$.

The result follows from Continuity Test using Sub-Basis.

$\blacksquare$