# if Ā = ∩ F then show that A ⊆ Ā and Ā is the smallest closed set that contains A?

hi

my question is:

The closure of A is denoted by Ā. Ā = ∩ F, where the intersection is taken

over all the closed sets F that contain A.

show:

1- A ⊆ Ā

2- Ā is the smallest closed set that contains A

### 3 Answers

- Anonymous8 years agoFavorite Answer
The first one on the question above is correct. A ⊆ Ā if for an arbitrary element a of A, a is also in Ā. If a is in A then it is in every set in F so it is in ∩ F = Ā.

The second one is not actually a proof. You proved that there exists a set B in F containing A, not that A is the smallest closed set that contains A.

2. Every closed set that contains A is in F (by definition of F). Ā is closed because it is the intersection of closed sets. Now assume there exists a closed set B smaller than Ā (B ⊂ Ā) containing A. You need to show that, in fact, B = Ā, a contradiction, which implies a smaller set cannot exist.

To show 2 sets are equal you need B ⊂ Ā and Ā ⊂ B. The first is true by assumption. The second is true because B is in F and Ā is in every set in F (because Ā = ∩ F). Hence B = Ā and Ā is the smallest closed set containing A.

- 8 years ago
1. Assume a is in A. Then a is in each F, so it is in the intersection of all the F. So, a is in Ā and hence A ⊆ Ā .

2. Let B be any closed set containing A. Then, B must be one of the F sets, so ∩ F ⊆B. Therefore, Ā ⊆B, and Ā is contained in every closed set that contains A. Finally, Ā itself is closed because it is the intersection of closed sets.