Real Analysis

Question 1

What makes a function ρ a metric?

Answer 1

1) ρ(x,y) ≥ 0 and equals 0 iff x=y
2) ρ(x,y)=ρ(y,x)
3) ρ(x,z) ≤ ρ(x,y) + ρ(y,z)

Question 2

What is “a set is open” equivalent to?

Answer 2

It is a union of balls

Question 3

What is the interior of A, int(A), defined as?

Answer 3

The largest open set contained in A

Question 4

What is int(A) equal to?

Answer 4

The union of all open sets in A

Question 5

When is x a limit/cluster/accumulation point of a set A? (x in X, A a subset of X)

Answer 5

If every ball about x contains a point of A other than x

Question 6

How is the set of limit points in A denoted?

Answer 6

A’

Question 7

What is the relation between A being closed and its limit points?

Answer 7

It is closed if it contains all of its limit points.

Question 8

A set A is closed if and only if?

Answer 8

Every convergent sequence in A has its limit in A

Question 9

If A is closed, what does this mean about its compliment?

Answer 9

The compliment of A is open

Question 10

What is the compliment of A? (X\A)

Answer 10

The set of all points in X which don’t belong to A

Question 11

What is the closure of A?

Answer 11

It is the smallest closed set containing A

Question 12

What is the closure of A, cl(A) equal to?

Answer 12

The union of A and A’, or the intersection of all the closed sets containing A.

Question 13

Which three statements are equivalent for a linear map T?

Answer 13

1) T is continuous
2) T is continuous at 0
3) T is bounded

Question 14

If a Cauchy sequence has a convergent subsequence, what does the subsequence converge to?

Answer 14

The same limit

Question 15

When is a metric space complete?

Answer 15

If every Cauchy sequence converges

Question 16

If (X,ρ) is complete and A is a subset of X, what is the completion of (A,ρ)?

Answer 16

(cl(A),ρ)

Question 17

What type of space is a complete normed linear space?

Answer 17

A Banach space

Question 18

How many fixed points does a contraction have and is it continuous?

Answer 18

A contraction is continuous and has at most one fixed point.

Question 19

What do we say about A if no disconnection of it exists?

Answer 19

It is connected

Question 20

Is the interval [0,1] connected?

Answer 20

Yes

Question 21

When is a set A clopen?

Answer 21

If it’s both closed and open

Question 22

What is the relationship between a subset A of X being connected, and clopeness?

Answer 22

A is connected if (A,ρ) has no non-trivial clopen set

Question 23

When is a subset K of X sequentially compact?

Answer 23

If every sequence in K has a convergent subsequence whose limit lies in K

Question 24

What is a cover called if every member of it is open?

Answer 24

An open cover

Question 25

What is a cover of K?

Answer 25

A collection of subsets for which K is the subset of the union of

Question 26

When is a subcollection of a cover called a subcover?

Answer 26

When it is also a cover of K, but smaller than the cover it is a subcollection of.

Question 27

When is K compact? (Heine Borel property)

Answer 27

If every open cover of K admits a finite subcover

Question 28

What is a finite cover?

Answer 28

A cover that has finitely many members

Question 29

In a metric space, what is sequential compactness equivalent to?

Answer 29

Compactness

Question 30

A compact set has what two properties?

Answer 30

It is closed and bounded

Question 31

What is a property of a closed subset of a compact set?

Answer 31

It is compact

Question 32

When can we say that (X,ρ) is a compact space?

Answer 32

If X is a compact set

Question 33

A set in R^n is totally bounded if and only if it is?

Answer 33

Bounded

Question 34

A metric space is compact if and only if?

Answer 34

It is complete and totally bounded

Question 35

A continuous function on a compact space is?

Answer 35

Uniformly continuous

Question 36

If T is a contraction on a complete space what does it have?

Answer 36

A unique fixed point