Definitions

Question 0

Interior point

Answer 0

Let E be a set of real numbers. A point x € E is said to be an interior point of E if it has a neighborhood which is contained in E.

Question 1

Neighborhood

Answer 1

A neighborhood of a point x is an open interval which contains x.

Question 2

Are all points in a open interval interior points?

Answer 2

Yes

Question 3

Are all points in a closed interval interior points?

Answer 3

No

Question 4

What are the interior points for the set of natural numbers?

Answer 4

There are none.

Question 5

What are the interior points for the set of rational numbers?

Answer 5

There are none.

Question 6

What are the interior points of the real numbers?

Answer 6

All of the reals.

Question 7

Isolated point

Answer 7

Let E be a set of real numbers. A point X € E is said to be an isolated point of E if there exists a neighborhood A of x so that A ^ E = {x}

Question 8

Are any points in a closed interval isolated points?

Answer 8

No. No points in a closed interval are isolated points.

Question 9

Are any points in an open interval isolated points?

Answer 9

No. No points in an open interval are isolated points.

Question 10

What are the isolated points for the set of natural numbers?

Answer 10

All of the natural numbers.

Question 11

What are the isolated points for the set of rational numbers?

Answer 11

There are none.

Between any two points there are infinitely many rational numbers.

Question 12

What are the isolated points for the set of real numbers?

Answer 12

There are none.

Question 13

Accumulation point

Answer 13

Let E be a set of real numbers. A point x is said to be an accumulation point for E if for every e > 0, the interval (x-e, x+e) contains infinitely many points in E.

Note that we do not require that the point x be in the set E.

Question 14

What are the accumulation points for the set of natural numbers?

Answer 14

There are none.

Question 15

What are the accumulation points for the set of rational numbers?

Answer 15

All of the real numbers.

Between every rational there is an irrational number.

Question 16

What are the accumulation points for the set of real numbers?

Answer 16

All of the real numbers.

Question 17

Boundary point

Answer 17

Let E be a set of real numbers. A point x is said to be a boundary point of E if for every e > 0, the interval (x-e, x+e) contains at least of point in E and at least one point in E complement.

Note that we do not require that the point x be in the set E.

Question 18

What are the boundary points in the set of natural numbers?

Answer 18

All of the natural numbers.

Question 19

What are the boundary points in the set of rational numbers?

Answer 19

All of the real numbers.

Question 20

What are the boundary points for the set of real numbers?

Answer 20

There are none.

Question 21

Must every interior point for a set E be an accumulation point for E?

Answer 21

Yes.

Question 22

Must every accumulation point for a set E be an interior point for the set E?

Answer 22

No. Not very accumulation point has to be an interior point for the set E.

Question 23

Is it possible for an accumulation point for a set E to be a boundary point?

Answer 23

Yes, it is possible.

Question 24

Must every isolated point for a set E be a boundary point for the set E?

Answer 24

Yes. Every isolated point for a set E must be a boundary point.

Question 25

Open set

Answer 25

A set E of real numbers is said to be open if every point of E is an interior point.

Question 26

Closed set (2 definitions)

Answer 26

> A set E is said to be closed if its complement is open.

> a set of real numbers F is closed if and only of every accumulation point of F converges to F.

Question 27

Compact (2 definitions)

Answer 27

> A set E of real numbers is said to be compact if every sequence {Xn} of values in E has a subsequence which converges to a point in the set E.
A set of real number is compact if and only if it is both closed and bounded.