A note on light induced mappings

Let a mapping f:X→Y between continua X and Y be given. We shall prove: a) if the induced mapping 2^{f}:2^{X}→2^{Y} is light, then w(X)=w(Y). In particular, if Y is metrizable, then X is metrizable, b) if the induced mapping C(f):C(X)→C(Y) is light and X is a D-continuum, then w(X)=w(Y).