X and y are sets zetau refers to the usual topology zetah

Question

X and Y are sets. ζ_u refers to the usual topology, ζ_H is the half-open interval topology. ζ_C is the open half-line topology, ζ_F 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, ζ_x) and (Y, ζ_y) 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.

Answer 1

These following questions are based on subsets of topology X and Y are sets. ?u refers to the normal topology, ?H is the half-open interval topology. ?C is the open half-line topology, ?F is the finite complement topology and ?cc. 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. i. If A and B are sets, then A ? B if and only if A n B = A. ii. Assume 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). iii. Assume 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). iv. Letf: X ? Y be a function and assume that A is a subset of Y. Then f(X- f-1 (Y - A)) ? A. v. Letf: X Y be a function and assume that A is a subset of Y. Then A ? f(X- f-1(Y - A)). vi. Let (X, ?) be a topological space and assume that A and B are subsets of X these that A ? B. Then Bd(A) ? Bd(B).