Exercise · ZW-TOPOLOGICAL-HOMEOMORPHISM-004 · ★☆☆ Basic
Let \(X=\{a,b\}\) with open sets \(\varnothing,\{a\},X\), and \(Y=\{0,1\}\) with open sets \(\varnothing,\{1\},Y\). Which bijections \(X\to Y\) are homeomorphisms ?
Choose one option.
A Only the one with \(f(a)=0\) B Both C Only the one with \(f(a)=1\) D Neither Show answer C. Only the one with \(f(a)=1\)← Back to the question
Explanation A homeomorphism must send the only proper open set \(\{a\}\) to the only proper open set \(\{1\}\), so \(f(a)=1\) and \(f(b)=0\). The other bijection sends \(\{a\}\) to \(\{0\}\), which is not open.
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 ZW-TOPOLOGICAL-HOMEOMORPHISM-001 If f:X→ Y is a homeomorphism, which statement is true?ZW-TOPOLOGICAL-HOMEOMORPHISM-002 Let X=\1,2\ and let id:(X,discrete)→(X,trivial). Is it a homeomorphism?ZW-TOPOLOGICAL-HOMEOMORPHISM-003 How many homeomorphisms are there from X=\1,2,3\ with the discrete topology to itself?ZW-TOPOLOGICAL-HOMEOMORPHISM-005 Let X=\1,2,3\ with open sets ∅,\1\,\1,2\,X, and Y=\a,b,c\ with open sets ∅,\c\,\b,c\,Y. If f:X→ Y i…ZW-TOPOLOGICAL-HOMEOMORPHISM-006 The function f:(0,1)→(0,∞), f(x)=(x)/(1-x), is a homeomorphism between intervals with the usual top…ZW-TOPOLOGICAL-HOMEOMORPHISM-007 Let id:(ℝ,discrete)→(ℝ,usual). Which statement is true?ZW-TOPOLOGICAL-HOMEOMORPHISM-008 How many homeomorphisms are there from the space X=\a,b\ with open sets ∅,\a\,X to itself?ZW-TOPOLOGICAL-HOMEOMORPHISM-009 Let X=\1,2,3\ with open sets ∅,\1\,\2,3\,X, and Y=\a,b,c\ with open sets ∅,\a\,\a,b\,Y. Both have 4…