Let X be a set with infinite c
Let X be a set with infinite cardinality. Define Tx ={U ⊆ X : U = Ø or X \ U is finite}. Prove that Tx is atopology on X. (Tx is called the Cofinite Topology orFinite Complement Topology.)
Answer:
be a set with infinite cardinality .
or \ 
is finite} .
Defination : A collection of subsets 
of is said to form a tpology on if it satisfies the following three conditions .
If 
be an finite collection then 
If 
be an infinite collection then 
Now we will prove that the collection or \ } is finite forms a topology on .
by defination of .
Also 
as \ 
is a finite set .
Suppose be a finite collection of elements of .
are all finite set .
is a finite set as finite union of finite set is finite.
is a finite set .
suppose be an infinite collection of elements of .
are all finite set .
is a finite set as arbitrary intersection of finite set is finite.
is a finite set .
So satisfies all three conditions to be a topology on 
Hence the collection or \ is finite } is a topology on
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