3.I.4F

Metric and Topological Spaces
Part IB, 2006

Which of the following are topological spaces? Justify your answers.

(i) The set X=ZX=\mathbf{Z} of the integers, with a subset AA of XX called "open" when AA is either finite or the whole set XX;

(ii) The set X=ZX=\mathbf{Z} of the integers, with a subset AA of XX called "open" when, for each element xAx \in A and every even integer n,x+nn, x+n is also in AA