141 Homeomorphism
A homeomorphism is a continuous bijection with continuous inverse that is used to identify topologically equivalent spaces.
Definition (Homeomorphism) A function between topological spaces \(f: \mathbf{U} \to \mathbf{V}\) is called a homeomorphism if the following conditions hold.
- \(f\) is a bijection and continuous
- \(f^{-1}\) is continuous
where
- \(f^{-1}\) is the inverse of \(f\)
- \(\mathbf{U}\), \(\mathbf{V}\) are topological spaces.
141.1 Elementary Example
141.1.1 Simple
A homeomorphism is a continuous bijection with continuous inverse.
\[ U = V = \{ 1,\ 2,\ 3 \} \]
\[ \tau = \{ \emptyset,\ \{1\},\ \{1,2\},\ U \} \]
\[ f(1)=1,\ f(2)=2,\ f(3)=3 \]
\[ f^{-1} = f \]
where
- \(f\) is a bijection.
- \(f\) and \(f^{-1}\) are continuous for this topology.
141.1.2 General
A relabeling of four points with matching topologies is a homeomorphism.
\[ U = \{ 1,\ 2,\ 3,\ 4 \},\quad V = \{ a,\ b,\ c,\ d \} \]
\[ f(1)=a,\ f(2)=b,\ f(3)=c,\ f(4)=d \]
\[ \tau_{U} = \{ \emptyset,\ \{1\},\ \{1,2\},\ U \},\quad \tau_{V} = \{ \emptyset,\ \{a\},\ \{a,b\},\ V \} \]
where
- \(f\) matches open sets of \(U\) with open sets of \(V\).
- \(f^{-1}\) is continuous as well.