description/proof of that for topological space and subsets, union of interiors of subsets is contained in interior of union of subsets
Topics
About: topological space
The table of contents of this article
Starting Context
Target Context
- The reader will have a description and a proof of the proposition that for any topological space and any subsets, the union of the interiors of the subsets is contained in the interior of the union of the subsets.
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:
\(T\): \(\in \{\text{ the topological spaces }\}\)
\(J\): \(\in \{\text{ the possibly uncountable index sets }\}\)
\(\{S_j \subseteq T \vert j \in J\}\):
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Statements:
\(\cup_{j \in J} Int (S_j) \subseteq Int (\cup_{j \in J} S_j)\)
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2: Proof
Whole Strategy: Step 1: apply the proposition that for any topological space and any subset, the interior of the subset is the set of the points of the space that have some open neighborhoods contained in the subset to see that \(\cup_{j \in J} Int (S_j) \subseteq Int (\cup_{j \in J} S_j)\).
Step 1:
Let \(t \in \cup_{j \in J} Int (S_j)\) be any.
\(t \in Int (S_j)\) for a \(j \in J\).
By the proposition that for any topological space and any subset, the interior of the subset is the set of the points of the space that have some open neighborhoods contained in the subset, \(t \in Int (S_j) = \{t \in T \vert \exists U_t \subseteq T \in \{\text{ the open neighborhoods of } t\} (U_t \subseteq S_j)\}\).
So, \(U_t \subseteq S_j\).
So, \(U_t \subseteq \cup_{j \in J} S_j\).
That means that \(t \in Int (\cup_{j \in J} S_j)\), by the proposition that for any topological space and any subset, the interior of the subset is the set of the points of the space that have some open neighborhoods contained in the subset.
So, \(\cup_{j \in J} Int (S_j) \subseteq Int (\cup_{j \in J} S_j)\).
