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

A continuous function $f$ between two topological spaces $X$ and $Y$ is generally defined as follows $f:X\to Y$ 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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Freddy Doyle
MathJax(?): Can't find handler for document MathJax(?): Can't find handler for document Here ${f}^{-1}\left(V\right)$ doesn't mean inverse function evaluated at $V$, but rather the pre-image of $V$ under $f$, that is

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
MathJax(?): Can't find handler for document MathJax(?): Can't find handler for document Instead of thinking of ${f}^{-1}\left(V\right)$ as it its inverse think about it as its preimage, that is:

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.