A continuous function f between two topological spaces X and Y is generally defined as follows f : X → Y is such that for every open subset V of Y, the set f − 1 ( V) is an open subset of X.In this definition only I have a confusion/doubt: it is not mentioned anywhere that f is bijective. So how can we take the inverse of f and define the continuity of f ? Moreover, there is homeomorphism in which it is necessary for f to be bijective and hence speaking of f − 1 , 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.

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A continuous function   f between two topological spaces   X and   Y is generally defined as follows   f  :  X  →  Y is such that for every open subset   V of   Y, the set       f          −      1      (  V) is an open subset of   X.In this definition only I have a confusion/doubt: it is not mentioned anywhere that   f is bijective. So how can we take the inverse of   f and define the continuity of   f ? Moreover, there is homeomorphism in which it is necessary for   f to be bijective and hence speaking of       f          −      1      , 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.

Freddy Doyle · Accepted Answer

MathJax(?): Can&#039;t find handler for document MathJax(?): Can&#039;t find handler for document Here       f          &amp;#x2212;      1        (  V  ) doesn&amp;#39;t mean inverse function evaluated at   V, but rather the pre-image of   V under   f, that is

2nalfq8 · Accepted Answer

MathJax(?): Can&#039;t find handler for document MathJax(?): Can&#039;t find handler for document Instead of thinking of       f          &amp;#x2212;      1        (  V  ) as it its inverse think about it as its preimage, that is:

A continuous function f between two topological spaces X and Y is generally define

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