Question

Prove or disprove that the intersection of any collection ofclosed sets is closed.

Answer 1

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 $B_i$ 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

$R =\bigcap\limits_{B_i \in F }^{} B_i$

Now, applying De-Morgan's Theorem, we have

$R^c = (\bigcap\limits_{B_i \in F }^{} B_i)^c$

$R^c = \bigcup\limits_{B_i \in F }^{} B_i^c$

Since we knew$B_i$ are closed set, thus, $B_i^c$ is an open set.

We also know that union of all open set is an open set.

Thus, $R^c$ is an open set.

Thus, R is a closed set.

Hence, the theorem.