Monday, August 5, 2019

functions - Union of preimages and preimage of union



Is it possible to have a map f:XY from a topological space X to a set Y and some subsets of Y namely Ui,iI such that iIf1(Ui) is not equal to f1(iIUi) ?




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:XY,
f1(Ui)=f1(Ui) and f1(Ui)=f1(Ui). Try proving this by elementary set
theory, i.e. take an element of f1(Ui) and show that it is an element of f1(Ui) and conversely.


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