Is it possible to have a map f:X→Y from a topological space X to a set Y and some subsets of Y namely Ui,i∈I such that ⋃i∈If−1(Ui) is not equal to f−1(⋃i∈IUi) ?
I can't think of a case that this is true, but of course this doesn't mean anything! Thanks.
Answer
It is a general fact that for any mapping of sets f:X→Y,
⋃f−1(Ui)=f−1(⋃Ui) and ⋂f−1(Ui)=f−1(⋂Ui). Try proving this by elementary set
theory, i.e. take an element of ⋃f−1(Ui) and show that it is an element of f−1(⋃Ui) and conversely.
No comments:
Post a Comment