description/proof of that for compact topological space and set of closed subsets closed under finite intersections, for open subset that contains intersection of set, there is element of set contained in open subset
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of compact topological space.
- The reader knows a definition of intersection of set.
Target Context
- The reader will have a description and a proof of the proposition that for any compact topological space and any set of some closed subsets closed under the finite intersections, for any open subset that contains the intersection of the set, there is an element of the set contained in the open subset.
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:
\(T\): \(\in \{\text{ the compact topological spaces }\}\)
\(S\): \(= \{C_j \in \{\text{ the closed subsets of } T\} \vert j \in J\}\) that is closed under the finite intersections of elements
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Statements:
\(\forall U \in \{\text{ the open subsets of } T\} \text{ such that } \cap S \subseteq U (\exists C_j \in S (C_j \subseteq U))\)
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2: Proof
Whole Strategy: Step 1: deal with the case that \(\emptyset \in S\), and suppose otherwise thereafter; Step 2: take a finite cover of \(T\), \(\{U\} \cup \{T \setminus C_j \vert j \in J^`\}\), and see that \(\cap_{j \in J^`} C_j \subseteq U\).
Step 1:
Let us suppose that \(\emptyset \in S\).
\(\emptyset \subseteq U\).
Let us suppose that \(\emptyset \notin S\), hereafter.
Step 2:
\(\{U\} \cup \{T \setminus C_j \vert j \in J\}\) is an open cover of \(T\), because for each \(t \in T\), when \(t \notin U\), \(t \notin \cap S\), because \(\cap S \subseteq U\), so, \(t \notin C_j\) for a \(j \in J\), so, \(t \in T \setminus C_j\) for that \(j\).
There is a finite subcover, \(\{U\} \cup \{T \setminus C_j \vert j \in J^`\}\), because \(T\) is compact.
\(\cap_{j \in J^`} C_j \in S\), because \(S\) is closed under the finite intersections of elements.
\(\cap_{j \in J^`} C_j \subseteq U\), because for each \(t \in \cap_{j \in J^`} C_j\), \(t \in C_j\) for each \(j \in J^`\), so, \(t \notin T \setminus C_j\) for each \(j \in J^`\), so, as \(\{U\} \cup \{T \setminus C_j \vert j \in J^`\}\) covers \(T\), \(t \in U\).
