5. Let (X,r) and (Y,A) be two topological spaces. Prove that a function f:(X.r)(Y.A) is closed if and only if (4)ss (4). VACX.
5. Let (X,r) and (Y,A) be two topological spaces. Prove that a function f:(X.r)(Y.A) is closed if and only if (4)ss (4). VACX.
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![5. Let (X,r) and (Y,A) be two topological spaces. Prove that a
function f:(X.r) Y,A) is closed if and only if
f(4)sf (A). VACX.
6. Consider the usual topological space (R.r,), where
Fu = { 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) and
Ix ={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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1ad816a5-be78-417f-80db-c365ea296a14%2F5583fa0c-cb57-49dd-bfb2-596a163650c0%2Fmxcgwl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:5. Let (X,r) and (Y,A) be two topological spaces. Prove that a
function f:(X.r) Y,A) is closed if and only if
f(4)sf (A). VACX.
6. Consider the usual topological space (R.r,), where
Fu = { 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) and
Ix ={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 4.1
1. Let (X,r) and (Y ,A) be two topological spaces. Prove that a
function f:(X,r)–(Y ,A) is continuous if and only if the
inverse 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 base
for A. Prove that a function f :(X,r)(Y,A) is continuous if
and only if the inverse image under of every member of ß is a
r- 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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1ad816a5-be78-417f-80db-c365ea296a14%2F5583fa0c-cb57-49dd-bfb2-596a163650c0%2Fycg03te_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Exercises 4.1
1. Let (X,r) and (Y ,A) be two topological spaces. Prove that a
function f:(X,r)–(Y ,A) is continuous if and only if the
inverse 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 base
for A. Prove that a function f :(X,r)(Y,A) is continuous if
and only if the inverse image under of every member of ß is a
r- 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.
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