I\'m working on an assignment in topology and I have come across this statement
ID: 2255434 • Letter: I
Question
I'm working on an assignment in topology and I have come across this statement in my textbook.
I do not understand X - X. Does this mean X is a complement of itself? Can you please explain? Thank you?
tonccuull Conslstng of X and only is also a topology on X; we shall call it the indiserete topology, or the trivial topology. MPLE 3. Let X be a set; let Tf be the collection of all subsets U of X such that X-U EXA either is finite or is all of X. Then Tf is a topology on X, called the finite complement topology. Both X and are in Tf, since an indexed family of nonempty elements of Tf, to show that UU, is in Tf, we compute X X is finite and X - is all of X. If (Ua) is The latter set is finite because each set X - Ua is finite. If U1, , Un are nonempty elements of Tf, to show that Ui is in Tf, we compute i-l The latter set is a finite union of finite sets and, therefore, finite.Explanation / Answer
Firstly X-X doesn't mean that X is a complement of itself it only means that complement of X is 'phi' whose cardinality is considered as 0, a finite number. So by definition 'phi' X belongs to the topology. Again X-'phi'=X i.e the complement of 'phi' is whole of X so again by definition 'phi' belongs to the topology. See the definition repeatedly it will be clear.