Exercise · ZW-TOPOLOGICAL-HOMEOMORPHISM-007 · ★★☆ Intermediate
Let \(\mathrm{id}:(\mathbb{R},\text{discrete})\to(\mathbb{R},\text{usual})\). Which statement is true?
Choose one option.
A It is a homeomorphism B It is not continuous C It is a continuous bijection, but not a homeomorphism D Its inverse is continuous, but it is not Show answer C. It is a continuous bijection, but not a homeomorphism← Back to the question
Explanation Every usual open set is open in the discrete topology , so the identity is continuous. Its inverse is not: \(\{0\}\) is open in the discrete topology and its preimage under the inverse is \(\{0\}\), which is not open in the usual topology . A continuous bijection is not enough to be a homeomorphism .
A bijection \(f\) between topological spaces \(X\) and \(Y\) that is continuous and whose inverse \(f^{-1}\) is also continuous.
\[f:X\to Y\ \text{bijective};\quad f,\ f^{-1}\ \text{continuous}\]
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