Products of Homeomorphic Spaces are Homeomorphic

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Theorem

Let:

$\sequence {T_i}_{i \mathop \in I}$
$\sequence {T'_i}_{i \mathop \in I}$

be indexed families of topological spaces, with indexing set $I$.

Let:

$\sequence {\phi_i}_{i \mathop \in I}$

be an indexed family of homeomorphisms $\phi_i$ from $T_i$ to $T'_i$.

Define:

$\ds T = \prod_{i \mathop \in I} T_i$
$\ds T' = \prod_{i \mathop \in I} T'_i$

where $\ds \prod_{i \mathop \in I} T_i$ denotes the product space.


Then, the mapping $\phi : T \to T'$ defined as:

$\map \phi x = \sequence {\map {\phi_i \circ \pr_i} x}_{i \mathop \in I}$

is a homeomorphism from $T$ to $T'$.


Proof

We have that:

$\map {\phi^{-1}} x = \sequence {\map {\phi_i^{-1} \circ \pr_i} x}_{i \mathop \in I}$

For:

\(\ds \map \phi {\map {\phi^{-1} } x}\) \(=\) \(\ds \sequence {\map {\phi_i \circ \pr_i} {\sequence {\map {\phi_i^{-1} \circ \pr_i} x}_{i \mathop \in I} } }_{i \mathop \in I}\)
\(\ds \) \(=\) \(\ds \sequence {\map {\phi_i \circ \phi_i^{-1} \circ \pr_i} x }_{i \mathop \in I}\)
\(\ds \) \(=\) \(\ds \sequence {\map {\pr_i} x}_{i \mathop \in I}\) Definition of Inverse Mapping
\(\ds \) \(=\) \(\ds x\)
\(\ds \map {\phi^{-1} } {\map \phi x}\) \(=\) \(\ds \sequence {\map {\phi_i^{-1} \circ \pr_i} {\sequence {\map {\phi_i \circ \pr_i} x}_{i \mathop \in I} } }_{i \mathop \in I}\)
\(\ds \) \(=\) \(\ds \sequence {\map {\phi_i^{-1} \circ \phi_i \circ \pr_i} x}_{i \mathop \in I}\)
\(\ds \) \(=\) \(\ds \sequence {\map {\pr_i} x}_{i \mathop \in I}\) Definition of Inverse Mapping
\(\ds \) \(=\) \(\ds x\)

It follows that $\phi$ is a bijection.

Additionally, by:

it follows that both $\phi$ and $\phi^{-1}$ are continuous.

Therefore, $\phi$ is a homeomorphism by definition.

$\blacksquare$

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