Assignment Help >> Mathematics
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.