Task 15. (0.3pt). Let x and y be topological spaces. A mapping f:x→y is said to be continuous if
for each open set UsubY the pre-image f1(u) is open subset in x.
Prove the following statements.
(a). The identity mapping 1x:x→x,1x(x)= x and the constant mapping c:x→ainY, taking x to an arbitrary point ainY are trivial examples of continuous functions.
(b) Let x is a discrete space and y is an arbitrary space. Then any function f:x→y is continuous.
(c) Let x is an arbitrary space, y is an antidiscrete space. Then any function f:x→y is continuous.