13: Euclidean Topological Space

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definition of Euclidean topological space

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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: Structured Description
• 2: Natural Language Description
• 3: Note

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Starting Context

• The reader knows a definition of topological space.
• The reader knows a definition of Euclidean topology.

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Target Context

• The reader will have a definition of Euclidean topological space.

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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: Structured Description

Here is the rules of Structured Description.

Entities:
$\ast {\mathbb{R}}^n$: with the Euclidean topology
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Conditions:
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2: Natural Language Description

Any ${\mathbb{R}}^n$ together with its Euclidean topology

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3: Note

It is important to be clear on these: ${\mathbb{R}}^n$ is just a set without any topology, which means that there is no proximity between points to say nothing of distance; Euclidean topological space does not imply any metric by itself, although many people say just "Euclidean" meaning having the Euclidean metric.

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References

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