Definitions Flashcards
Interior point
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.
Neighborhood
A neighborhood of a point x is an open interval which contains x.
Are all points in a open interval interior points?
Yes
Are all points in a closed interval interior points?
No
What are the interior points for the set of natural numbers?
There are none.
What are the interior points for the set of rational numbers?
There are none.
What are the interior points of the real numbers?
All of the reals.
Isolated point
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}
Are any points in a closed interval isolated points?
No. No points in a closed interval are isolated points.
Are any points in an open interval isolated points?
No. No points in an open interval are isolated points.
What are the isolated points for the set of natural numbers?
All of the natural numbers.
What are the isolated points for the set of rational numbers?
There are none.
Between any two points there are infinitely many rational numbers.
What are the isolated points for the set of real numbers?
There are none.
Accumulation point
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.
What are the accumulation points for the set of natural numbers?
There are none.
What are the accumulation points for the set of rational numbers?
All of the real numbers.
Between every rational there is an irrational number.
What are the accumulation points for the set of real numbers?
All of the real numbers.
Boundary point
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.
What are the boundary points in the set of natural numbers?
All of the natural numbers.
What are the boundary points in the set of rational numbers?
All of the real numbers.
What are the boundary points for the set of real numbers?
There are none.
Must every interior point for a set E be an accumulation point for E?
Yes.
Must every accumulation point for a set E be an interior point for the set E?
No. Not very accumulation point has to be an interior point for the set E.
Is it possible for an accumulation point for a set E to be a boundary point?
Yes, it is possible.
Must every isolated point for a set E be a boundary point for the set E?
Yes. Every isolated point for a set E must be a boundary point.
Open set
A set E of real numbers is said to be open if every point of E is an interior point.
Closed set (2 definitions)
> 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.
Compact (2 definitions)
> 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.
