description/proof of that for topological space and finite number of subsets, interior of intersection of subsets is intersection of interiors 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 finite number of subsets, the interior of the intersection of the subsets is the intersection of the interiors 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:
\(T\): \(\in \{\text{ the topological spaces }\}\)
\(J\): \(\in \{\text{ the finite index sets }\}\)
\(\{S_j \subseteq T \vert j \in J\}\):
//
Statements:
\(Int (\cap_{j \in J} S_j) = \cap_{j \in J} Int (S_j)\)
//
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 \(Int (\cap_{j \in J} S_j) = \{t \in T \vert \exists U_t \subseteq T \in \{\text{ the open neighborhoods of } t\} (\forall j \in J (U_t \subseteq S_j))\}\); Step 2: see that \(\{t \in T \vert \exists U_t \subseteq T \in \{\text{ the open neighborhoods of } t\} (\forall j \in J (U_t \subseteq S_j))\} = \cap_{j \in J} \{t \in T \vert \exists U_t \subseteq T \in \{\text{ the open neighborhoods of } t\} (U_t \subseteq S_j)\}\).
Step 1:
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, \(Int (\cap_{j \in J} S_j) = \{t \in T \vert \exists U_t \subseteq T \in \{\text{ the open neighborhoods of } t\} (U_t \subseteq \cap_{j \in J} S_j)\}\).
But \(U_t \subseteq \cap_{j \in J} S_j\) equals \(\forall j \in J (U_t \subseteq S_j)\).
So, \(Int (\cap_{j \in J} S_j) = \{t \in T \vert \exists U_t \subseteq T \in \{\text{ the open neighborhoods of } t\} (\forall j \in J (U_t \subseteq S_j))\}\).
Step 2:
Let us see that \(\{t \in T \vert \exists U_t \subseteq T \in \{\text{ the open neighborhoods of } t\} (\forall j \in J (U_t \subseteq S_j))\} = \cap_{j \in J} \{t \in T \vert \exists U_t \subseteq T \in \{\text{ the open neighborhoods of } t\} (U_t \subseteq S_j)\}\).
For each \(t\) in the left hand side, for each \(j \in J\), \(t \in \{t \in T \vert \exists U_t \subseteq T \in \{\text{ the open neighborhoods of } t\} (U_t \subseteq S_j)\}\), so, \(t\) is in the right hand side.
For each \(t\) in the right hand side, for each \(j \in J\), \(t \in \{t \in T \vert \exists U_t \subseteq T \in \{\text{ the open neighborhoods of } t\} (U_t \subseteq S_j)\}\), so, let such a \(U_t\) be denoted as \(U_{t, j}\), because it depends on \(j\), then, \(\cap_{j \in J} U_{t, j}\) is an open neighborhood of \(t\), because \(J\) is finite, and for each \(j \in J\), \(\cap_{j \in J} U_{t, j} \subseteq U_{t, j} \subseteq S_j\), so, \(t\) is in the left hand side.
So, \(\{t \in T \vert \exists U_t \subseteq T \in \{\text{ the open neighborhoods of } t\} (\forall j \in J (U_t \subseteq S_j))\} = \cap_{j \in J} \{t \in T \vert \exists U_t \subseteq T \in \{\text{ the open neighborhoods of } t\} (U_t \subseteq S_j)\} = \cap_{j \in J} Int (S_j)\).
So, \(Int (\cap_{j \in J} S_j) = \cap_{j \in J} Int (S_j)\).
