Thread: Open and Closed set topology question

1. Open and Closed set topology question

Let be a collection of subset of the set X, satisfying:

i. and X contained in ,
ii. finite unions of elements in are in , and
iii. arbitrary unions of elements in are in

We know that is a topology on X

Give an example of a topology when X = . What are the open sets in this topology?What are the closed sets? What is the basis for the topology?

2. Originally Posted by flaming
Let be a collection of subset of the set X, satisfying:

i. and X contained in ,
ii. finite unions of elements in are in , and
iii. arbitrary unions (intersections?) of elements in are in

We know that is a topology on X
I assume you mean arbitrary intersections because both "finite unions" and "arbitrary unions" conditions at the same time do not make much sense.

Give an example of a topology when X = .
Cofinite topology on R

What are the open sets in this topology?What are the closed sets?
Finite point sets are closed in the cofinite topology on $\mathbb{R}$.

What is the basis for the topology?
A subbasis of a cofinite topology on R is the set of complements of singletons in R. You can get the basis for a cofinite topology on R using the subbasis.