Exercise · ZW-TOPOLOGICAL-HOMEOMORPHISM-008 · ★★☆ Intermediate
How many homeomorphisms are there from the space \(X=\{a,b\}\) with open sets \(\varnothing,\{a\},X\) to itself?
Choose one option.
Show answer
- A. 1
Explanation
Only the identity: a homeomorphism must preserve the only proper open set \(\{a\}\), so it fixes \(a\) and hence \(b\). The swap \(a\leftrightarrow b\) sends \(\{a\}\) to \(\{b\}\), which is not open.
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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