description/proof of that for topological space and finite number of open covers, intersection of covers is open cover
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of topological space.
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.
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.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\(T\): \(\in \{\text{ the topological spaces }\}\)
\(A_1\): \(\in \{\text{ the possibly uncountable index sets }\}\)
...
\(A_n\): \(\in \{\text{ the possibly uncountable index sets }\}\)
\(A\): \(= A_1 \times ... \times A_n\)
\(C_1\): \(= \{U_{1, \alpha_1} \vert \alpha_1 \in A_1\}\), \(\in \{\text{ the open covers of } T\}\)
...
\(C_n\): \(= \{U_{n, \alpha_n} \vert \alpha_n \in A_n\}\), \(\in \{\text{ the open covers of } T\}\)
\(C\): \(= \{U_{1, \alpha_1} \cap ... \cap U_{n, \alpha_n} \vert \alpha = (\alpha_1, ..., \alpha_n) \in A\}\): 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 \(\emptyset\) 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 \(T\).
