Exercise · ZW-CONTINUOUS-IMAGE-OF-CONNECTED-SPACE-003 · ★☆☆ Basic
If \(X\) is connected and \(f:X\to Y\) is continuous, what can be guaranteed about \(f(X)\)?
Choose one option.
Show answer
- A. It is connected
Explanation
The theorem only guarantees connectedness. Open, closed or compact depend on \(f\) and \(Y\); for instance \(x\mapsto x^2\) sends \(\mathbb R\) to \([0,\infty)\), which is not compact.
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
- ZW-CONTINUOUS-IMAGE-OF-CONNECTED-SPACE-001 Let f:[-1,2]→mathbb R, f(x)=x^2. The image f([-1,2]) is:
- ZW-CONTINUOUS-IMAGE-OF-CONNECTED-SPACE-002 Which function f:[-1,1]→mathbb R cannot be continuous, because its image is not connected?
- 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…
- ZW-CONTINUOUS-IMAGE-OF-CONNECTED-SPACE-005 For all these topologies on Y=\a,b,c\, except one, there is a continuous surjection from some conne…
- 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:
- ZW-CONTINUOUS-IMAGE-OF-CONNECTED-SPACE-009 Onto which of these subspaces of mathbb R is there no continuous surjection from mathbb R?
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