Question

1) Let Y=[-1,1]have the standard topology.
Which of the following sets are open in Y and which are open in R?
A=(-1,-1/2)∪(1/2,1)
B=(-1,-1/2]∪[1/2,1)
C=[-1,-1/2)∪(1/2,1]
D=[-1,-1/2]∪[1/2,1]

2) Let Y=(0,5]. In each case in the set open, closed, both or neither in Y in the standard topology?
1/(0,1) 2/(0,1] 3/{1} 4/(0,5] 5/(1,2) 6/[1,2) 7/(1,2] 8/[1,2] 9/(4,5] 10/[4,5].

3) Let X be a Housdorff topological space, and Y a subset of X. Prove that the subspace topology in Y is a Housdorrf.

4) Let X be a topological space and let Y⊂X have subspace topology.
a) If A is open in Y, and Y is open in X, show that A is open in X.
b) if A is closed in Y and Y is closed in X show that A is closed in X.

5) Show the standard topology on Q, the set of rational numbers, is not the discrete topology.

6) Is the finite complement topology on R² the same as the product topology on R² that results from taking the product R_fc×R_fc, where R_fc. Is R the finite complement topology? Justify your answer?

7) Let X =PPR²(0,0), the particular point on R² with the origin serving as the particular point. Is X the same as the topology that results from taking the product of R itself, where each R has a particular point topology PPR 0? Justify your answer.

8) Let S² be the sphere, D be the disk, T be the torus, S^1 be the circle and I=[0,1] with standard topology. Draw pictures of the product spaces S×I, T×I, S^1×I×I, and S^1×D.

9) If L is line in the plane describe the subspace topology it inherits from Rl×R and from Rl×Rl. Where Rl is the real line in the lower limit topology. Note that results depend on the slope of the line. In all cases, it is a familiar topology.

10) Show that If A⊂X and B⊂Y then Cl(A×B)=Cl(A)×Cl(B).

11) Determine whether or not the sets in are open, closed, both or neither in the product typologies on the plane given by R×R, Rl×R,and Rl×Rl. Where Rl is the real line in the lower limit topology. See Question.jpg.

12) Suppose that A⊂X and B⊂Y
Provide an example demonstrating that ∂(A×B)=∂(A)×∂(B) does not hold in general.
Derive and prove relationship expressing ∂(A×B) in terms of ∂(A), ∂(B), A and B.
Topology Exercises

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1)
A=(-1,-1/2)∪(1/2,1) is open in Y and open in R.
B=(-1,-1/2]∪[1/2,1) is not open in Y and in R is not open and not closed.
C=[-1,-1/2)∪(1/2,1] i is open in Y and in R is not open and not closed.
D=[-1,-1/2]∪[1/2,1] is closed in Y and closed in R....

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