8: Topology

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definition of topology

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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

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Starting Context

• The reader knows a definition of set.

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Target Context

• The reader will have a definition of topology.

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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:
$S$: $\in \{\text{ the sets }\}$
$\ast O$: $\subseteq Pow(S)$
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Conditions:
1) $\mathrm{\varnothing }\in O$ and $S\in O$
$\wedge$
2) for any $U_1\in O$ and any $U_2\in O$, $U_1\cap U_2\in O$
$\wedge$
3) for any $U_j\in O$ where $j\in J$ where $J$ is any index set not necessarily countable, $({\cup }_{j\in J}{U}_j)\in O$
//

Each element of $O$ is called "open subset".

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References

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