Exercise · ZW-TOPOLOGICAL-HOMEOMORPHISM-006 · ★★☆ Intermediate
The function \(f:(0,1)\to(0,\infty)\), \(f(x)=\dfrac{x}{1-x}\), is a homeomorphism between intervals with the usual topology . What is \(f^{-1}(y)\)?
Choose one option.
A \(\dfrac{y}{1-y}\) B \(\dfrac{1}{1+y}\) C \(1+y\) D \(\dfrac{y}{1+y}\) Show answer ← Back to the question
Explanation Solve: \(y=\frac{x}{1-x}\Rightarrow y-yx=x\Rightarrow x=\frac{y}{1+y}\). This formula is continuous on \((0,\infty)\) and takes values in \((0,1)\), so \(f\) is a bijection with continuous inverse .
A bijection \(f\) between topological spaces \(X\) and \(Y\) that is continuous and whose inverse \(f^{-1}\) is also continuous.
\[f:X\to Y\ \text{bijective};\quad f,\ f^{-1}\ \text{continuous}\]
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