2.4 Homeomorphisms
Here we will define a homeomorphism, which is a special type of continuous functions on a topological
space. Some examples of homeomorphisms will also be presented.
Definition 2.4.1. Let \(X\) and \(Y\) be topological spaces. A function \(f:X\rightarrow Y\) is said to be a homeomorphism if the following conditions are satisfied,
- i
- \(f\) is both injective and surjective,
- ii
- \(f:X\rightarrow Y\) and its inverse \(f^{-1}:Y\rightarrow X\) are both continuous.
Two topological spaces \(X\) and \(Y\) are said to be homeomorphic if there exists a homeomorphism \(f:X\rightarrow Y\) from \(X\) to
\(Y\).
Example 2.4.2. Any two open intervals \((a,b)\) and \((c,d)\) in \(\mathbb {R}\) are homeomorphic. The function \(f:(a,b)\rightarrow (c,d)\) defined by
\(f(x)=c+\dfrac {(d-c)(x-a)}{(b-a)}\) is a homeomorphism.
Example 2.4.3. Let \(X=\{1,2,3,4\}\), \(Y=\{\alpha ,\beta ,\gamma ,\delta \}\) and define \(\tau _X=\{\{4\},\emptyset ,X\}\) and
\(\tau _Y=\{\{\beta \},Y,\emptyset \}\). The function \(f:X\rightarrow Y\) defined by \(f(1)=\delta \), \(f(2)=\alpha \), \(f(3)=\gamma \), \(f(4)=\beta \) is a homeomorphism.
\(f^{-1}(\emptyset _Y)=\emptyset _X\), \(f^{-1}(Y)=X\), \(f^{-1}(\{\beta \})=\{4\}\)
\(f^{-1}:Y\rightarrow X\); \(f^{-1}(\alpha )=2\), \(f^{-1}(\beta )=4\), \(f^{-1}(\gamma )=3\), \(f^{-1}(\delta )=1\)
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