231: Euclidean Topological Space Is 2nd Countable

<The previous article in this series | The table of contents of this series | The next article in this series>

A description/proof of that Euclidean topological space is 2nd countable

---

Topics

About: topological space

---

The table of contents of this article

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

---

Starting Context

• The reader knows a definition of Euclidean topological space.
• The reader knows a definition of 2nd countable topological space.
• The reader admits the proposition that for any Euclidean topological space, the set of all the open balls with rational centers and rational radii is a basis.
• The reader admits the proposition that any subset of any countable set is countable.
• The reader admits the proposition that the product of any finite number of countable sets is countable.

---

Target Context

• The reader will have a description and a proof of the proposition that any Euclidean topological space is 2nd countable.

---

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.

---

Main Body

---

1: Description

Any Euclidean topological space, ${\mathbb{R}}^n$, is 2nd countable.

---

2: Proof

By the proposition that for any Euclidean topological space, the set of all the open balls with rational centers and rational radii is a basis, ${\mathbb{R}}^n$ has the set of all the open balls with rational centers and rational radii as a basis. The basis is countable, because the elements can be indexed by (the center coordinates, the radius) pairs while the rational numbers set is countable and 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 hold true.

---

References

<The previous article in this series | The table of contents of this series | The next article in this series>