Exercise · ZW-TOPOLOGICAL-HOMEOMORPHISM-010 · ★★★ Advanced
Let \(X=\{1,2,3,4\}\) with open sets \(\varnothing,\{1\},\{2\},\{1,2\},X\). How many homeomorphisms \(X\to X\) are there?
Choose one option.
Show answer
- C. 4
Explanation
A homeomorphism must permute the minimal open sets \(\{1\}\) and \(\{2\}\) among themselves (2 ways) and, since \(3\) and \(4\) lie in exactly the same open sets, it may swap them freely (2 ways): \(2\cdot2=4\). Not all \(24\) bijections work.
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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