Reference no: EM132761209

Question 1. Show that if Y is a subspace of X, and A is a subset of Y, then the topology A

Question 2. Show that if A is closed in Y and Y is closed in X, then A is closed in X.

Question 3. Show that if A is closed in X and B is closed in Y, then A x B is closed in X x Y.

Question 4. Show that if U is open in X and A is closed in X, then U - A is open in X, and A U is closed in X.

Question 5. If Τ and Τ' are topologies on X and Τ' is strictly finer than Τ, what can you say about the corresponding subspace topologies on the subset Y of X?

Question 6. A map f:X→Y is said to be an open map if for every open set U of X, the set f (U) is open in Y. Show that Π₂:  X x Y → Y are open maps.

Question 7. In general, let us denote the identity function for a set C by ic. That is, define ic: C → C to be the function given by the rule ic(x) = x for all x ∈ C.

 Given f : A → B, we say that a function g : B → A is a left inverse for f if gof = iA; and we say that h : B → A is aright inverse for f if foh = i_B.

a) Show that if f has a left inverse, f is injective; and if f has a right inverse, f is surjective.

b) Give an example of a function that has a left inverse but no right inverse.

Question 8. Let A, B, and A_α denote subsets of a space X. Prove the following.

(a) If A ⊂ B, then A¯ ⊂ B¯.

(b) (AUB)¯ =A¯U B¯.

(c) U A_α ⊃ U Aα¯ give an example where equality fails.

Question 9.  Criticize the following "proof" that UAα¯ ⊂ UA¯a: if (Aa) is a collection of sets in X and if x ∈ UAa¯, then every neighborhood U of x intersects UAα. Thus U must intersect some Aα, so that x must belong to the closure of some Aα,. Therefore, x ∈ U A¯a.

Question 10. Let f : A → B. Let A0 ⊂ A and Bo ⊂ B.

(a) Show that A0 ⊂ f^-1 (f (A0) and that equality holds if f is injective.

(b) Show that f ( f^-1 (B0)) ⊂ Bo and that equality holds if f is surjective.

Question 11. Prove that for functions f. R → R, the ∈-δ definition of continuity implies the open set definition.

Question 12. X and X' denote a single set in the two topologies T and T', respectivly

Let i : X' → X be the identity function.

(a) Show that i is continuous <=>T' is finer than T.

(b) Show that i is a homeomorphism <=>T' = T.