# Set Theory - Question 4.

?
What is a boundary point, b(A)?
The set of points which do not belong to the interior or exterior of A.
1 of 8
What is an exterior point, ext(A)?
The interior complement of A, ie. int(A‹).
2 of 8
What can the closure of A be written as?
The union of the interior and boundary of A, ie. Ā = A° U b(A).
3 of 8
What is a convergent sequence?
The sequence <a1, a2, …> of real numbers converges to b e R; if for every € > 0, there exists a positive integer n• such that n > n• implies )an - b( < €.
4 of 8
What is the intermediate value theorem?
If a function is continuous on a closed interval and u is a value between f(a) and f(b), then there exists c e [a,b] such that f(c) = u.
5 of 8
What is the extreme value theorem?
In every interval [a,b] where a function is continuous, there is at least one maximum and one minimum; ie. it has at least two extreme values.
6 of 8
Prove that f is continuous iff. for every open set V, f^-1 is open.
Suppose f is continuous on A. f^-1 is open in A. But A is open in R, and f^-1(V) = f^-1(V) ∏ A. So f^-1 is open.
7 of 8
Proving backwards...
If f^-1(V) is open in R, then f^-1 ∏ A is open in A. f is defined from A to R, so preimage of any set is contained in A, so f^-1(V) ∏ A = f^-1(V). So f^-1(V) is open.
8 of 8

## Other cards in this set

### Card 2

#### Front

What is an exterior point, ext(A)?

#### Back

The interior complement of A, ie. int(A‹).

### Card 3

#### Front

What can the closure of A be written as?

#### Back ### Card 4

#### Front

What is a convergent sequence?

#### Back ### Card 5

#### Front

What is the intermediate value theorem?

#### Back 