Respuesta :
Answer:
Intersection of collection of any closed set is a closed set.
Step-by-step explanation:
Let F be a collection of arbitrary closed sets and let [tex]B_i[/tex] be closed set belonging to F.
We define a closed set as the set that contains its limit point or in other words it can be described that the complement or not of a closed set is an open set.
Thus, we can write R as
[tex]R =\bigcap\limits_{B_i \in F }^{} B_i[/tex]
Now, applying De-Morgan's Theorem, we have
[tex]R^c = (\bigcap\limits_{B_i \in F }^{} B_i)^c[/tex]
[tex]R^c = \bigcup\limits_{B_i \in F }^{} B_i^c[/tex]
Since we knew[tex]B_i[/tex] are closed set, thus, [tex]B_i^c[/tex] is an open set.
We also know that union of all open set is an open set.
Thus, [tex]R^c[/tex] is an open set.
Thus, R is a closed set.
Hence, the theorem.
