220: Neighborhoods Basis at Point on Topological Space

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definition of neighborhoods basis at 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 neighborhood of point on topological space.

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Target Context

• The reader will have a definition of neighborhoods basis at 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, $O$
$t$: $\in T$
$\ast B_t$: $\subseteq \{\text{ the neighborhoods of }t\text{ on }T\}$
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Conditions:
$\mathrm{\forall }{N}_t^{\prime }\in \{\text{ the neighborhoods of }t\text{ on }T\}(\mathrm{\exists }{N}_t\in B_t(N_t\subseteq N_t^{\prime }))$
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2: Note

$B_t$ does not need to be a set of open neighborhoods of $t$ although it may be a set of open neighborhoods of $t$.

When $B_t$ is a set of open neighborhoods of $t$, it is called "open neighborhoods basis at $t$ on $T$".

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References

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