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jlk
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Urysohn's Metrization Theorem states that every Hausdorff second-countable regular space is metrizable.

What is an example of a Hausdorff second-countable regular space where it is difficult to prove metrizability without using Urysohn's Theorem?

For example, the theorem applies toimplies that a (second-countable) manifold is metrizable. However, this result can be proven without using Urysohn's Theorem by provingshowing directly that every such manifold embeds in $\mathbb{R}^{\infty}$ (using partitions of unity).

Urysohn's Metrization Theorem states that every Hausdorff second-countable regular space is metrizable.

What is an example of a Hausdorff second-countable regular space where it is difficult to prove metrizability without using Urysohn's Theorem?

For example, the theorem applies to a (second-countable) manifold. However, this can be proven without using Urysohn's Theorem by proving directly that every such manifold embeds in $\mathbb{R}^{\infty}$ (using partitions of unity).

Urysohn's Metrization Theorem states that every Hausdorff second-countable regular space is metrizable.

What is an example of a Hausdorff second-countable regular space where it is difficult to prove metrizability without using Urysohn's Theorem?

For example, the theorem implies that a (second-countable) manifold is metrizable. However, this result can be proven without using Urysohn's Theorem by showing directly that every such manifold embeds in $\mathbb{R}^{\infty}$ (using partitions of unity).

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jlk
jlk
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  • 33
  • 48

What is a good application of Urysohn's Theorem?

Urysohn's Metrization Theorem states that every Hausdorff second-countable regular space is metrizable.

What is an example of a Hausdorff second-countable regular space where it is difficult to prove metrizability without using Urysohn's Theorem?

For example, the theorem applies to a (second-countable) manifold. However, this can be proven without using Urysohn's Theorem by proving directly that every such manifold embeds in $\mathbb{R}^{\infty}$ (using partitions of unity).