17: Basis of Topological Space

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definition of basis of 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 neighborhood of point.

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Target Context

• The reader will have a definition of basis of 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:
$T$: $\in \{\text{ the topological spaces }\}$
$\ast B$: $\subseteq Pow(T)$
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Conditions:
$\mathrm{\forall }U\in B(U\in \{\text{ the open subsets of }T\})$
$\wedge$
$\mathrm{\forall }t\in T,\mathrm{\forall }{N}_t\subseteq T\in \{\text{ the neighborhoods of }t\}(\mathrm{\exists }U\in B(t\in U\subseteq N_t))$
//

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2: Natural Language Description

For any topological space, $T$, any set of some open sets such that for each neighborhood, $N_t$, of each point, $t\in T$, there is an open set, $U\in B$, such that $t\in U\subseteq N_t$

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3: Note

When the topology of $T$ is not determined yet, any basis determines the topology, by the proposition that any basis of any topological space determines the topology.

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References

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