Exercise · ZW-CONTINUOUS-IMAGE-OF-CONNECTED-SPACE-009 · ★★★ Advanced
Onto which of these subspaces of \(\mathbb R\) is there no continuous surjection from \(\mathbb R\)?
Choose one option.
Show answer
- D. \(\mathbb R\setminus\{0\}\)
Explanation
\(\mathbb R\setminus\{0\}\) is not connected. The others are images of \(\mathbb R\): for example via \(\sin^2 x\), \(1/(1+e^x)\) and \(x^2\).
Review the concept: Continuous image of a connected space
If \(f:X\to Y\) is a continuous function between topological spaces and \(X\) is connected, then the image \(f(X)\) is a connected subspace of \(Y\).
\[X\text{ connected},\ f\text{ continuous}\ \Longrightarrow\ f(X)\text{ connected}\]
More practice problems: continuous image of a connected space
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- ZW-CONTINUOUS-IMAGE-OF-CONNECTED-SPACE-004 Let X=\1,2,3\ with open sets ∅,\1\,\1,2\,X and Y=\a,b\ with open sets ∅,\a\,Y. How many continuous…
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- ZW-CONTINUOUS-IMAGE-OF-CONNECTED-SPACE-006 Which is a counterexample to «if f is continuous and f(X) is connected, then X is connected»?
- ZW-CONTINUOUS-IMAGE-OF-CONNECTED-SPACE-007 Let f:[a,b]→mathbb R be continuous with f(a)=-2 and f(b)=3. Which value must f attain?
- ZW-CONTINUOUS-IMAGE-OF-CONNECTED-SPACE-008 The image of f:mathbb R→mathbb R, f(x)=(1)/(1+x^2), is:
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