Countable Infinite Product of Real Number Spaces is Homeomorphic to Fréchet Metric Space
Jump to navigation
Jump to search
Theorem
Let $\struct {\R, \tau_d}$ denote the real number line under the Euclidean topology.
Let $T = \struct {\R^\omega, \tau} = \ds \prod_{i \mathop \in \N} \struct {\R, \tau_d}$ denote the countable-dimensional real Cartesian space under the product topology $\tau$.
Let $\struct {\R^\omega, d}$ be the Fréchet space on $\R^\omega$, where:
- $\map d {x, y} = \ds \sum_{i \mathop \in \N} \dfrac {2^{-i} \size {x_i - y_i} } {1 + \size {x_i - y_i} }$
Then the topology induced by $d$ is exactly the product topology $\tau$.
Proof
This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $37$. Fréchet Space: $7$