Exercise · ZW-TOPOLOGICAL-HOMEOMORPHISM-001 · ★☆☆ Basic
If \(f:X\to Y\) is a homeomorphism, which statement is true?
Choose one option.
Show answer
- B. \(U\) is open in \(X\) if and only if \(f(U)\) is open in \(Y\)
Explanation
Since \(f\) and \(f^{-1}\) are continuous, preimages of open sets are open in both directions, so \(U\) is open exactly when \(f(U)\) is. Distances need not be preserved (a homeomorphism may stretch), and \(X\), \(Y\) can be different sets.
Review the concept: Homeomorphism
A bijection \(f\) between topological spaces \(X\) and \(Y\) that is continuous and whose inverse \(f^{-1}\) is also continuous.
\[f:X\to Y\ \text{bijective};\quad f,\ f^{-1}\ \text{continuous}\]
More practice problems: homeomorphism
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