Cartesian Product of Homeomorphisms is Homeomorphism
Jump to navigation
Jump to search
Theorem
Let $S_1, S_2, T_1, T_2$ be topological spaces.
Let $f_1: S_1 \to T_1$ and $f_2: S_2 \to T_2$ be mappings.
Let:
- $f_1 \times f_2: S_1 \times S_2 \to T_1 \times T_2$
be defined as:
- $\forall \tuple {x, y} \in S_1 \times S_2: \map {\paren {f_1 \times f_2} } {x, y} = \tuple {\map {f_1} x, \map {f_2} y}$
where $S_1 \times S_2$ denotes the product space of $S_1$ and $S_2$, and similarly for $T_1 \times T_2$.
Let $f_1$ and $f_2$ be homeomorphisms.
Then $f_1 \times f_2$ is also a homeomorphism.
Proof
From Cartesian Product of Mappings is Continuous iff Factor Mappings are Continuous:
- $f_1 \times f_2$ is continuous.
From Cartesian Product of Bijections is Bijection:
- $f_1 \times f_2$ is a bijection.
From Cartesian Product of Mappings is Continuous iff Factor Mappings are Continuous:
- $\paren {f_1 \times f_2}^{-1} = f_1^{-1} \times f_2^{-1}$ is continuous.
So, by definition, $f_1 \times f_2$ is a homeomorphism.
$\blacksquare$
Sources
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $3$: Continuity generalized: topological spaces: Exercise $3.9: 20$