In this definition only I have a confusion/doubt: it is not mentioned anywhere that is bijective. So how can we take the inverse of and define the continuity of ? Moreover, there is homeomorphism in which it is necessary for to be bijective and hence speaking of , in this regard, makes sense.
Also, measurable functions in measure theory are defined in a similar fashion to that of continuous functions in topology. Here also I am confused about the same thing! Please give some insights. Thank you in advance.