804: For Topological Space and Finite Number of Open Covers, Intersection of Covers Is Open Cover

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description/proof of that for topological space and finite number of open covers, intersection of covers is open cover

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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: Proof

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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 description and a proof of the proposition that for any topological space and any finite number of any open covers, the intersection of the covers is an open cover.

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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:
$T$: $\in \{\text{ the topological spaces }\}$
$A_1$: $\in \{\text{ the possibly uncountable index sets }\}$
...
$A_n$: $\in \{\text{ the possibly uncountable index sets }\}$
$A$: $=A_1\times ...\times A_n$
$C_1$: $=\{U_{1,{\alpha }_1}|{\alpha }_1\in A_1\}$, $\in \{\text{ the open covers of }T\}$
...
$C_n$: $=\{U_{n,{\alpha }_n}|{\alpha }_n\in A_n\}$, $\in \{\text{ the open covers of }T\}$
$C$: $=\{U_{1,{\alpha }_1}\cap ...\cap U_{n,{\alpha }_n}|\alpha =({\alpha }_1,...,{\alpha }_n)\in A\}$: which is being called "the intersection of the open covers"
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Statements:
$C\in \{\text{ the open covers of }T\}$
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2: Proof

Whole Strategy: Step 1: take each $t\in T$ and see that $t$ is contained in an element of $C$: Step 2: see that each element of $C$ is open on $T$; Step 3: conclude the proposition.

Step 1:

Let $t\in T$ be any.

Let us see that $t$ is contained in an element of $C$.

As $C_j$ is a cover, there is a $U_{j,{\beta }_j}\in C_j$ such that $t\in U_{j,{\beta }_j}$.

$\beta =({\beta }_1,...,{\beta }_n)\in A$, and so, $U_{1,{\beta }_1}\cap ...\cap U_{n,{\beta }_n}\in C$.

$t\in U_{1,{\beta }_1}\cap ...\cap U_{n,{\beta }_n}$.

Step 2:

Each element of $C$ is $U_{1,{\alpha }_1}\cap ...\cap U_{n,{\alpha }_n}$, which is open on $T$, as a finite intersection of open subsets.

Step 3:

There may be some duplications in $C$, but such duplications are automatically eliminated by the definition of set (no set has any duplication). And $\mathrm{\varnothing }$ may be contained in $C$, but that is not any problem for $C$'s being an open cover.

So, $C$ is an open cover of $T$.

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References

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