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

Lucian Maddox 2022-07-08 Answered
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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Answers (2)

Freddy Doyle
Answered 2022-07-09 Author has 20 answers
MathJax(?): Can't find handler for document MathJax(?): Can't find handler for document Here f 1 ( V ) doesn't mean inverse function evaluated at V, but rather the pre-image of V under f, that is
f 1 ( V ) = { x X f ( x ) V } .
You are correct in saying that f need not be bijective in these definitions, but that isn't necessary for what the notation means.
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2nalfq8
Answered 2022-07-10 Author has 3 answers
MathJax(?): Can't find handler for document MathJax(?): Can't find handler for document Instead of thinking of f 1 ( V ) as it its inverse think about it as its preimage, that is:
f 1 ( V ) := { x X : f ( x ) V }
If f is a bijection there is a chance for it to be a homeomorphism.
In the case of measurable functions, it is very similar: A function is measurable if given a measurable set in the codomain then its preimage is measurable.
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