Exercise · ZW-CONTINUOUS-IMAGE-OF-CONNECTED-SPACE-006 · ★★☆ Intermediate
Which is a counterexample to «if \(f\) is continuous and \(f(X)\) is connected, then \(X\) is connected»?
Choose one option.
A \(X=\{1,2\}\) discrete and \(f\) constant B \(X=\{1,2\}\) indiscrete and \(f\) constant C \(X=[0,1]\) and \(f\) the identity D \(X=\mathbb R\) and \(f(x)=x\) Show answer A. \(X=\{1,2\}\) discrete and \(f\) constant← Back to the question
Explanation A counterexample needs a disconnected \(X\). Only the two-point discrete space is; the constant map is continuous and its image, a point, is connected.
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