- #1
Bacle
- 662
- 1
Hi,
Let f :X-->Y ; X,Y topological spaces is any map and {Ui: i in I} is a cover for X
so that :
f|_Ui is continuous, i.e., the restriction of f to each Ui is continuous, then:
1) If I is finite , and the {Ui} are all open (all closed) , we can show f is continuous:
taking W open in Y, f^-1(W)= \/ f^-1(W /\ Ui) = is the union of open sets in X;
each W/\Ui is open, and W/\Ui is contained in Ui.
( where \/ is the union over I ; /\ is intersection over I ); same for V closed in Y.
2) If I is infinite, the argument can break down , i.e., if {Ui } is a closed cover
for X (each Ui is closed) ; f|_Ui is continuous and I is infinite, then this result
fails. Does anyone know of an example of this last?
Let f :X-->Y ; X,Y topological spaces is any map and {Ui: i in I} is a cover for X
so that :
f|_Ui is continuous, i.e., the restriction of f to each Ui is continuous, then:
1) If I is finite , and the {Ui} are all open (all closed) , we can show f is continuous:
taking W open in Y, f^-1(W)= \/ f^-1(W /\ Ui) = is the union of open sets in X;
each W/\Ui is open, and W/\Ui is contained in Ui.
( where \/ is the union over I ; /\ is intersection over I ); same for V closed in Y.
2) If I is infinite, the argument can break down , i.e., if {Ui } is a closed cover
for X (each Ui is closed) ; f|_Ui is continuous and I is infinite, then this result
fails. Does anyone know of an example of this last?