11: Neighborhood of Point on Topological Space

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definition of neighborhood of point on 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: Note

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Starting Context

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

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Target Context

• The reader will have a definition of neighborhood of point on 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:
$T$: $\in \{\text{ the topological spaces }\}$, with any topology, $O$
$t$: $\in T$
$\ast N_t$: $\subseteq T$
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Conditions:
$\mathrm{\exists }{U}_t\in O(t\in U_t\subseteq N_t)$
//

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2: Note

$N_t$ does not need to be any open subset, although it needs to contain an open subset.

When $N_t$ is open, it is called "open neighborhood of $t$ on $T$": $U_t$ is an open neighborhood of $t$ on $T$.

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. But in some cases, whether it is a (not-necessarily-open) neighborhood or an open neighborhood matters: for example, when there is a compact neighborhood of a point, there may not be any compact open neighborhood of the point.

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References

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