15: Locally Topologically Euclidean Topological Space

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A definition of locally topologically 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: Definition
• 2: Note

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Starting Context

• The reader knows a definition of topological space.
• The reader knows a definition of neighborhood of point.
• The reader knows a definition of homeomorphism.
• The reader knows a definition of Euclidean topological space.

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Target Context

• The reader will have a definition of locally topologically 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: Definition

For any Euclidean topological space, ${\mathbb{R}}^n$, any topological space, $T$, such that at its any point, $p\in T$, there are an open neighborhood, $U_p\subseteq T$, of $p$, and an open neighborhood, $U_q\subseteq {\mathbb{R}}^n$, of a point, $q\in {\mathbb{R}}^n$, such that there is a homeomorphism, $f:U_p\to U_q$

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2: Note

The expression, "topologically Euclidean topological space", may seem redundant, but is not so strictly speaking, because 'Riemannianly Euclidean topological space' is possible because any Riemannian manifold is a topological space, as well as 'only topologically Euclidean Riemannian manifold' is of course possible.

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References

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