Exercise · ZW-TOPOLOGICAL-HOMEOMORPHISM-002 · ★☆☆ Basic
Let \(X=\{1,2\}\) and let \(\mathrm{id}:(X,\text{discrete})\to(X,\text{trivial})\). Is it a homeomorphism ?
Choose one option.
A Yes: it is a continuous bijection B No: it is bijective, but neither it nor its inverse is continuous C No: it is not bijective D No: it is bijective and continuous, but its inverse is not continuous Show answer D. No: it is bijective and continuous, but its inverse is not continuous← Back to the question
Explanation The identity is bijective. It is continuous because every preimage is a subset and in the discrete topology every subset is open. Its inverse goes from the trivial to the discrete space and is not continuous: \(\{1\}\) is open in the target but its preimage \(\{1\}\) is not open in the trivial topology.
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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