Exercise · ZW-TOPOLOGICAL-HOMEOMORPHISM-003 · ★☆☆ Basic
How many homeomorphisms are there from \(X=\{1,2,3\}\) with the discrete topology to itself?
Choose one option.
Show answer
- A. 6
Explanation
In the discrete topology every bijection is continuous, and so is its inverse. The homeomorphisms are therefore all the bijections: \(3!=6\).
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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