Continuous Image of Connected Space is Connected/Corollary 2
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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.
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.
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $6.2$: Connectedness: Corollary $6.2.13$