Chapter 1&2

Question 1

Topological space

Answer 1

The set X together with a topology T on X

Question 2

Trivial topology

Answer 2

T={empty set, X}

Question 3

Discrete Topology

Answer 3

T= collection of all subsets of X

Question 4

Finite Complement Topology

Answer 4

T = empty set and every set in R with a finite complement

Question 5

Finer

Coarser

Answer 5

If T_1 is a subset of T_2 then T_2 is ______ than T_1 and T_1 is _____ than T_2

Question 6

A topology T on A set X is…

Answer 6

A collection of subsets of X, (open sets) such that:

1) empty set and X are open sets
2) the intersection of finitely many open sets is an open set
3) the Union of any collection of open sets is an open set

Question 7

Neighborhood of x

Answer 7

Let X be a topological space and x in X. An open set U containing x is said to be a…

Question 8

Thm 1.4

Answer 8

Let X be a topological space and let A be a subset of X. Then A is open if and only if for each x in A, there is a neighborhood U of x such that x in U is a subset of A.

Question 9

Basis for a topology on X

Answer 9

1) for each x in X there is a B in ‘B’ such that x in B
2) if B_1 and B_2 are in ‘B’ and x in B_1 intersect B_2, then there exists B_3 in ‘B’ such that x in B_3 subset B_1 intersect B_2

Question 10

The topology T generated by a basis, ‘B’, on a set X…

Answer 10

Is generated by defining the open sets to be the empty set and every set that is equal to a union of basis elements

Question 11

Standard topology

Answer 11

Topology generated by ‘B’ ={(a,b) subset of R | a<b></b>

Question 12

Let ‘B’ be a basis. Assume that B_1,…,B_n in ‘B’ and that x in intersection _i=1 ^n B_i. Then…

Answer 12

There exists B’ in ‘B’ such that x in ‘B’ subset intersection _i=1 ^n B_i

Question 13

Lower limit topology

Answer 13

Generated by ‘B’={[a,b) subset R|a<b></b>

Question 14

Upper limit topology

Answer 14

Generated by ‘B’ ={(a,b] subset R| a<b></b>

Question 15

Digital line topology

Answer 15

B(n) = {n} if n is odd
{n-1, n, n+1} if n is even

Generated by
‘B’={B(n)| n in Z}

Question 16

Let X be a set and ‘B’ be a basis for a topology on X. Then U is open in the topology generated by ‘B’ if and only if

Answer 16

For each x in U there exists a basis element B_x in ‘B’ such that x in B_x subset U

Question 17

Basis for a topology on R^2

Answer 17

Collection ‘B’={B(x, e)|x in R^2, e>0}

Question 18

Let y be in R^2 and assume r>0. Then for every x in B(y,r)…

Answer 18

There exists an e>0 such that B(x, e) subset B(y, r)

Question 19

Basis for standard topology on R^2

Answer 19

‘B’={(a,b)x(c,d) subset R^2|a<b></b>

Question 20

Let X be a set with topology T, and let C be a collection of open sets in X. If, for each open set U in X and for each x in U, there is

Answer 20

A basis that generates the topology T

Question 21

Vertical line topology

Answer 21

Topology generated by ‘B’={{a}x(b,c) subset R^2 | a,b,c in R}

Question 22

Closed

Answer 22

Subset A of a topological space X is ____ if the set X-A is open

Question 23

The closed ball of radius e centered at x

Answer 23

For each x in R^2 and e>0. B(x,e)={y in R^2| d(x,y) less than or equal to e}

Question 24

Closed rectangle

Answer 24

If [a,b] and [c,d] are closed bounded intervals in R, then the product [a,b]x[c,d] subset R^2 is called

Question 25

D(x,y)

Answer 25

Euclidean distance between x and y

Question 26

Closed balls and closed rectangles are…

Answer 26

Closed sets in the standard topology on R^2

Question 27

Let X be a topological space. The following statements about collection of closed sets in X hold:

Answer 27

1) empty set and X are closed
2) the intersection of any collection of closed sets is a closed set
3) the Union of finitely many closed sets is a closed set

Question 28

Hausdorff

Answer 28

If for every pair of distinct points x and y in X, there exists disjoint neighborhoods U and V of x and y respectively

Question 29

If X is a Hausdorff space, then….

Answer 29

Every single-point subset of X is closed

Question 30

Infinite comb

Answer 30

Subset of the plane defined by C={(x,0)|0 leq x leq 1} U {(1/2^n,y)| n=0,1,2,… and 0 leq y leq 1}

Question 31

Let A be a subset of topological space X. The interior of A is

Answer 31

The union of all open sets contained in A.

Question 32

Let A be a subset of a topological space X. The closure of A

Answer 32

Is the intersection of all closed sets containing A

Question 33

Let X be a topological space and A and B be subsets of X.

If U is an open set in X and U subset A

Answer 33

Then U subset Int(A)

Question 34

Let X be a topological space and A and B be subsets of X.

If C is a closed set in X and A subset C

Answer 34

Then Cl(A) subset C

Question 35

Let X be a topological space and A and B be subsets of X.

If A subset B then

Answer 35

Int(A) subset Int(A)

Question 36

Let X be a topological space and A and B be subsets of X.

If A subset B then

Answer 36

Cl(A) subset Cl(B)

Question 37

Let X be a topological space and A and B be subsets of X.

A is open if and only if

Answer 37

A=Int(A)

Question 38

Let X be a topological space and A and B be subsets of X.

A is closed if and only if

Answer 38

A=Cl(A)

Question 39

Dense

Answer 39

If Cl(A) = X for a subset of a topological space X then it is called

Question 40

Let X be a topological space, A be a subset of X, and y be an element of X. Then y in Int(A) if and only if

Answer 40

There exists an open set U such that y in U subset A

Question 41

Let X be a topological space, A be a subset of X, and y be an element of X. Then y in Cl(A) if and only if

Answer 41

Every open set containing y intersects A

Question 42

For sets A and B in a topological space X, the following statements hold:

Answer 42

1) Int(X-A)=X - Cl(A)
2) Cl(X-A)= X - Int(A)
3) Int(A)UInt(B) subset Int(AUB)
4) Int(A) intersect Int(B) = Int(A intersect B)

Question 43

Limit point

Answer 43

Let A be a subset of a topological space X. A point x in X is a ____ of A if every neighborhood of x intersects A in a point other than x.

Question 44

Let A be a subset of a topological space X, and let A’ be the set of limit points of A. Then…

Answer 44

Cl(A)=A U A’

Question 45

A subset A of a topological space is closed if and only if

Answer 45

It contains all of its limit points

Question 46

In a topological space X, a sequence (x_1,x_2,…) converges to x in C if

Answer 46

For every neighborhood U of x, there is a positive integer N such that x_n in U for all n geq N.

Question 47

Let A be a subset of R^n in the standard topology. If x is a limit point of A, then…

Answer 47

There is a sequence of points in A that converges to x

Question 48

If X is a Hausdorff space then

Answer 48

Every convergent sequence of points in X converges to a unique point in X

Question 49

Let A be a subset of a topological space X. The boundary of A is

Answer 49

Cl(A)-Int(A)

Question 50

Let A be a subset of a topological space X and let x be a point in X. Then x in boundary of A if and only if

Answer 50

Every neighborhood of x intersects both A and X-A

Question 51

Let A be a subset of a topological space X. Then the following statements about the boundary of A hold:

Answer 51

1) boundary of A is closed
2) boundary of A = Cl(A) intersect Cl(X-A)
3) boundary of A intersect Int(A) = empty set
4) boundary of A U Int(A) = Cl(A)
5) boundary of A subset A if and only if A is open
6) boundary A intersect A = empty set if and only if A is open
7) boundary of A = empty set if and only if A is both open and closed