230: For Euclidean Topological Space, Set of All Open Balls with Rational Centers and Rational Radii Is Basis

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A description/proof of that for Euclidean topological space, set of all open balls with rational centers and rational radii is basis

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Topics

About: topological space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Description
• 2: Proof
• 3: Note

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Starting Context

• The reader knows a definition of Euclidean topological space.
• The reader knows a definition of basis of topological space.
• The reader admits the proposition that any open set on any Euclidean topological space has a rational point.

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Target Context

• The reader will have a description and a proof of the proposition that for any Euclidean topological space, the set of all the open balls with rational centers and rational radii is a basis.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Description

For any Euclidean topological space, ${\mathbb{R}}^n$, the set of all the open balls with rational centers and rational radii, $B_{rat}$, is a basis.

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2: Proof

Let us take any open set, $U\subseteq {\mathbb{R}}^n$. By the proposition that any open set on any Euclidean topological space has a rational point, there is a rational point, $p\in U$. By the definition of Euclidean topological space, there is an open ball, $p\in B_{p-ϵ}\subseteq U$. But there is a rational number, $0<{ϵ}^{\prime }\le ϵ$, because the infinite decimal of $ϵ$ can be truncated at a digit. So, there is the open ball, $B_{p-{ϵ}^{\prime }}\in B_{rat}$, such that $B_{p-{ϵ}^{\prime }}\subseteq U$. By the definition of basis of topological space, $B_{rat}$ is a basis.

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3: Note

$B_{rat}$ is countable, as the rational numbers set is countable, by the proposition that any subset of any countable set is countable and the proposition that the product of any finite number of countable sets is countable.

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References

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