I need help to solve this exercise please? 5. Let (X,r) and (Y,A) be two topological spaces. Prove that afunction f:(X.r) Y,A) is closed if and only iff(4)sf (A). VACX.6. Consider the usual topological space (R.r,), whereFu = { V cR:Vx eV 3(a,b)cR such that x e (a,b)cr }U{}Let (X,r) be topological space. Where: X (-1,1) andIx ={V cX :Vx eV 3(a,b)cX such that x e (a,b)cV }U{®}Show that (X.r,) =(R,r, ).7. Show that the boundedness is not a topological property. Exercises 4.11. Let (X,r) and (Y ,A) be two topological spaces. Prove that afunction f:(X,r)–(Y ,A) is continuous if and only if theinverse image under r of every A-closed set is a r-closed set.2. Let (X ,r) and (Y ,A) be two topological spaces. Let p be a basefor A. Prove that a function f :(X,r)(Y,A) is continuous ifand only if the inverse image under of every member of ß is ar- open set.3. If f :(X.r)(Y.A) and g:(Y,A)-(Z.r) are open functions,prove that gof :(X.r)(Z.r) is open function.4. If f:(X,r)-(Y,A) and g:(Y.A) (Z.r)are closed functions,prove that gof :(X.r)(Z.r) is closed function.

Name: I need help to solve this exercise please? 5. Let (X,r) and (Y,A) be t
Uploaded: 2024-03-20T07:07:37.000Z
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Description: I need help to solve this exercise please? 5. Let (X,r) and (Y,A) be two topological spaces. Prove that afunction f:(X.r) Y,A) is closed if and only iff(4)sf (A). VACX.6. Consider the usual topological space (R.r,), whereFu = { V cR:Vx eV 3(a,b)cR su