Question

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

Answer 1

Intersection of collection of any closed set is a closed set.

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.