10. Let X be a compact space and let Y be a Hausdorff space. Prove that X the projection Ty : X × Y → Y defined by Ty(x, y) = y is a closed continuous function.

Let (¥,r) and (¥ ,4) be two topological spaces. Prove that a function f :(X .r)——(¥ ,A) is continuous if and only if the inverse image under / of every A- closed set is a r- closed set.

Q4) Consider the space (X.T) and let A = [0,2) U (3) U (4,7). Find d(A), A, Aº, ext(A) and b(A).

Find the image of the closed disk |z| ≤ 1 under the  given linear mapping w = f(z) and represent the linear mapping with a sequence of plots such as  translation, magnification, rotation. *f(z)=3iz

Let X and Y be normed spaces and F:X→Y (a) be a continuous linear map. Show that F is closed . Is the converse true ? Justify.

Let (X, T) and (Y, T1) be two topological spaces and let f be a continuous mapping of X into Y. If (Y, T1) is a T1 space, then (X, T) is a T1 space?

6. (i) Let f (X, J) → (Y, U), where (X, J) and (Y, U) are two topolog- ical spaces. Show that if (X, T) is a discrete space, then f is continuous. (ii) Let X = {a,b,c}. construct two topologies T₁ and J₂ on X. Then define a function f (X, J₁)→ (X, J₂) so that f is continuous. :

Find the f(a)- n(x²_ 2y+y@) definition set and drawan the plane

Let f(x,y,z,w)=45x3z-1/y+wz3. Evaluate f(a,b,c,d) where a,b,c are non-zerol real numbers

The largest set on which the function f(x, y) = 1/(2 - x² - y²) is continuous is A. The exterior of the circle x² + y² = 2 B. All of the xy-plane except the circle x² + y² = 2 C. The interior of the circle x² + y² = 2, plus the circle D. The interior of the circle x² + y² = 2 E. All of the xy-plane