101: Subspace Topology of Subset of Topological Space

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definition of subspace topology of subset 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.

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Target Context

• The reader will have a definition of subspace topology of subset 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:
$T^{\prime }$: $\in \{\text{ the topological spaces }\}$ with any topology, $O^{\prime }$
$T$: $\subseteq T^{\prime }$
$\ast O$: $\in \{\text{ the topologies of }T\}$, $=\{U^{\prime }\cap T|U^{\prime }\in O^{\prime }\}$
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Conditions:
//

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2: Natural Language Description

For any topological space, $T^{\prime }$, with any topology, $O^{\prime }$, and any subset, $T\subseteq T^{\prime }$, the topology of $T$, $O=\{U^{\prime }\cap T|U^{\prime }\in O^{\prime }\}$

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3: Note

$O$ is indeed a topology of $T$: 1) $T\in O$, because $T=T^{\prime }\cap T$ and $T^{\prime }\in O^{\prime }$; $\mathrm{\varnothing }\in O$, because $\mathrm{\varnothing }=\mathrm{\varnothing }\cap T$ and $\mathrm{\varnothing }\in O^{\prime }$; 2) for any possibly uncountable number of elements of $O$, $\{U_{\alpha }\in O|\alpha \in A\}$, ${\cup }_{\alpha \in A}{U}_{\alpha }\in O$, because ${\cup }_{\alpha \in A}{U}_{\alpha }={\cup }_{\alpha \in A}(U_{\alpha }^{\prime }\cap T)=({\cup }_{\alpha \in A}{U}_{\alpha }^{\prime })\cap T$, by the proposition that for any set, the intersection of the union of any possibly uncountable number of subsets and any subset is the union of the intersections of each of the subsets and the latter subset, and ${\cup }_{\alpha \in A}{U}_{\alpha }^{\prime }\in O^{\prime }$; 3) for any finite number of elements of $O$, $\{U_j\in O|j\in J\}$, ${\cap }_{j\in J}{U}_j\in O$, because ${\cap }_{j\in J}{U}_j={\cap }_{j\in J}(U_j^{\prime }\cap T)=({\cap }_{j\in J}{U}_j^{\prime })\cap T$, by the proposition that for any set, the intersection of the intersection of any possibly uncountable number of subsets and any subset is the intersection of the intersections of each of the subsets and the latter subset, and ${\cap }_{j\in J}{U}_j^{\prime }\in O^{\prime }$.

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References

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