11: Neighborhood of Point

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definition of neighborhood of point

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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 Definition
• 3: Note

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Starting Context

• The reader knows a definition of topological space.
• The reader knows a definition of open set.

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Target Context

• The reader will have a definition of neighborhood of point.

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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:
$T$: $\in \{\text{ the topological spaces }\}$
$p$: $\in T$
$\ast N_p$: $:\subseteq T$
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Conditions:
$\mathrm{\exists }{U}_p\in \{\text{ the open subsets of }T\}(p\in U_p\subseteq N_p)$
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2: Natural Language Definition

For any topological space, $T$, and its any point, $p\in T$, any subset, $N_p\subseteq T$, that contains an open set, $U_p\subseteq T$, that contains $p$, which is $p\in U_p\subseteq N_p\subseteq T$

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3: Note

Some people mean 'open neighborhood' by 'neighborhood', while in many cases, whether it is a (not-necessarily-open) neighborhood or an open neighborhood does not matter, because, for example, "there is a neighborhood" equals "there is an open neighborhood", because if there is a neighborhood, there is an open neighborhood in it, and if there is an open neighborhood, there is the neighborhood that is the open neighborhood.

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References

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