Exercise · ZW-CONTINUOUS-IMAGE-OF-A-COMPACT-SPACE-002 · ★☆☆ Basic
If \(f:[0,1]\to\mathbb{R}\) is continuous, which of these cannot be \(f([0,1])\)?
Choose one option.
Show answer
- A. \((0,1)\)
Explanation
The image must be compact. \((0,1)\) is not (it is not closed); the other three are compact and arise from continuous functions.
Review the concept: Continuous image of a compact space
If \(f:X\to Y\) is continuous and the space \(X\) is compact, then the image \(f(X)\) is a compact subspace of \(Y\).
\[f(X)\ \text{compact}\]
More practice problems: continuous image of a compact space
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