A description/proof of that intersection or finite union of closed sets is closed
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of closed set.
- The reader admits the proposition that for any set, the intersection of the compliments of any possibly uncountable number of subsets is the complement of the union of the subsets.
- The reader admits the proposition that for any set, the union of the complements of any possibly uncountable number of subsets is the complement of the intersection of the subsets.
Target Context
- The reader will have a description and a proof of the proposition that the intersection of any possibly uncountable number of closed sets or the union of any finite number of closed sets is closed.
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: Description
For any topological space, T, the intersection of any possibly uncountable number of closed sets, \(\cap_\alpha C_\alpha\) where \(C_\alpha \subseteq T\) closed where \({\alpha}\) is any possibly uncountable indices set, or the union of any finite number of closed sets, \(\cup_i C_i\) where \(C_i \subseteq T\) closed where \({i}\) is any finite indices set, is closed.
2: Proof
The complement of the intersection, \(T \setminus \cap_\alpha C_\alpha\), equals \(T \setminus \cap_\alpha (T \setminus U_\alpha)\) where \(U_\alpha := T \setminus C_\alpha\) open. But \(\cap_\alpha (T \setminus U_\alpha) = T \setminus \cup_\alpha U_\alpha\) by the proposition for any set, the intersection of the compliments of any possibly uncountable number of subsets is the complement of the union of the subsets. As \(\cup_\alpha U_\alpha\) is open, \(T \setminus \cup_\alpha U_\alpha\) is closed, so, \(T \setminus \cap_\alpha C_\alpha\) is open.
The complement of the union, \(T \setminus \cup_i C_i\), equals \(T \setminus \cup_i (T \setminus U_i)\) where \(U_i := T \setminus C_i\) open. But \(\cup_i (T \setminus U_i) = T \setminus \cap_i U_i\) by the proposition that for any set, the union of the complements of any subsets is the complement of the intersection of the subsets. As \(\cap_i U_i\) is open, \(T \setminus \cap_i U_i\) is closed, so, \(T \setminus \cup_i C_i\) is open.
