Continuous Image of Connected Space is Connected/Corollary 2

Corollary to Continuous Image of Connected Space is Connected

Let $T$ be a connected topological space.

Let $f: T \to \R$ be a continuous real-valued mapping.

Then $f \sqbrk T$ is a real interval.

Proof

From Continuous Image of Connected Space is Connected, the continuous image of the connected space $T$ is connected.

The result follows from Subset of Real Numbers is Interval iff Connected.

$\blacksquare$

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $6.2$: Connectedness: Corollary $6.2.13$