Openness and monotoneity of induced mappings

Abstract

It is shown that for locally connected continuum X if the induced mapping C(f) : C(X) --> C(Y) is open,then f is monotone. As a corollary it follows that if the continuum X is hereditarily locally connected and C(f) is open, then f is a homeomorphism. An example is given to show that local connectedness is essential in the result.