+1-415-670-9189
info@expertsmind.com
Finite complement topology
Course:- Mathematics
Reference No.:- EM13891864




Assignment Help
Assignment Help >> Mathematics

X and Y are sets. ζu refers to the usual topology, ζH is the half-open interval topology. ζC is the open half-line topology, ζF is the finite complement topology and ζcc. is the countable complement topology. R is the set of real numbers, Z is the set of integers and Q is the set of rational numbers.

1. If A and B are sets, then A ⊆ B if and only if A ∩ B = A.

2. Suppose that A and B are subsets of X and C and D are subsets of Y.

Then (A x C) (B x D) = (A - B) x (C - D).

3. Suppose that A and B are subsets of X and C and D are subsets of Y.

Then (A x C) U (B x D) ⊆ (A U B) x (C U D).

4. Letf : X → Y be a function and suppose that A is a subset of Y.

Then f(X- f-1 (Y - A)) ⊆ A.

5. Letf : X Y be a function and suppose that A is a subset of Y.

Then A ⊆ f(X- f-1(Y - A)) .

6. Let (X, ζ) be a topological space and suppose that A and B are subsets of X such that A ⊆ B.

Then Bd(A) ⊆ Bd(B).

7. Let (X, ζ) be a topological space and suppose that A and B are subsets of X such that A ⊆ B.

Then Int(A)  ⊆ Int(B).

8. Let (X, ζ) be a topological space and suppose that A and B are subsets of X such that A ⊆ B.

Then C1(A) ⊆ Cl (B).

9. Let (X, ζ) be a topological space and suppose that A and B are subsets of X such that A ⊆ B.

Then A' ⊆ B'

10. Suppose that (X, ζx) and (Y, ζy) are topological spaces and f : X → 0 Y is a function. If f is continuous then, for every closed set A ⊆ Y, f-1 (A) is closed.

11. Suppose that (X, ζx) and (Y, ζy) are topological spaces and f : X → Y is a function. If for every closed set A  Y, f 1(A) is closed then f is continuous.

12. Suppose that (X, ζx) and (Y, 3y) are topological spaces andf : X → Y is a function . If f is continuous then, for every closed set A ⊆ X, f(A) is closed.

13. Suppose that (X, ζx) and (Y, 3y) are topological spaces andf : X → Y is a function . If, for every closed set A ⊆ X, f(A) is closed then f is continuous.

14. Let ζ = {(-∞ , n) : n ∈ U {R, Ø}. Then ζ is a topology on R.

15. Let ζ = {(-∞,x) : x  ∈ Q} U {R, Ø} . Then ζ is a topology on R.




Put your comment
 
Minimize


Ask Question & Get Answers from Experts
Browse some more (Mathematics) Materials
Determine whether the statement is true or false. If it is true, explain why it is true. If false, give an example to show it is false.
Your rich aunt gives you a high school graduation present of $110,000. Her present is in the form of a 7-year bond with an annual interest rate of 8.4%. The bond says that i
Jack and Jill ran a 100 meter race. Jill ran the race in 10 seconds and won by 10 meters; Jack had run only 90 meters when Jill crossed the finish line.
It costs a $15 flat fee to rent a chain saw, plus $3 a day starting with the first day. Let x represent the number of days rented, so y represents the charge to the user (in
Beth, Bill, and Eddie together work 6 hours. Then, Bill and Eddie do it in 9. Beth, Bill, and Eddie work 3 hours together. Beth and Bill spent 6 hours more. How long would i
In polar coordinates, write equations for the line x=1 and the circle of radius 2 centered at the origin. Write an integral in polar coordinates representing the area of the r
There are 560 third- and fourth-grade students in King Elementary School. If there are 80 more third- graders than fourth-graders, how many third- graders are there in the s
Compute the state transition matrix for system - Determine the variation of the following functional and calculate the Hessian and show that it is singular at the origin.