Finite Complement Topology: Why It's the Finest

[beetle2]

Hi Guys
I was wondering if anyone knows of a good link that shows why the finite complement is a Topology?
I been told it is the finest topology is this right?

 

[Bacle]

Beetle2 (Is that Paul?. Ringo.?. John.?George.?)
" Hi Guys
I was wondering if anyone knows of a good link that shows why the finite complement is a Topology? I been told it is the finest topology is this right? "

I think it is a good, tho maybe a little tedious exercise in point-set topology:

1)Let A1,A2 be in cofinite, so that X-A1, X-A2 is finite. Using DeMorgan:

(X-A1)\/(X-A2)=X-(A1/\A2) . Can you see why X-( X-(A1/\A2)) is in (X,T), or why


any union of sets in (X,T) is in (X,T).?


2) (X-A1)/\(X-A2)=(X-(A1\/A2)). What happens with X-(X-(A1\/A2)).? (Can you see

why you need finitely-many Xn's here.?

 

[Bacle]

Forgot to add a few things:

I think you need to add some conditions to state whether a topology is finest

or not (maybe with respect to some map being continuous, e.g.)

Depending on your definition of finer topology ( unfortunately, I have seen that

different people mean opposite things by this term), the finest topology in a

space X is given by 2^X --the discrete topology--and the coarser one is given

by (X, empty.)

 

[beetle2]

1)Let A1,A2 be in cofinite, so that X-A1, X-A2 is finite. Using DeMorgan:

(X-A1)\/(X-A2)=X-(A1/\A2) . Can you see why X-( X-(A1/\A2)) is in (X,T)


Is it because X-( X-(A1/\A2)) = A1/\A2 and the intercection of any closed sets in T1 is a finite set.

 

[Werg22]

It's straightforward, just remember De Morgan's laws for sets.