# A Co-finite topology on $X$ is a discrete topology on $X$?(If $X$ is a finite set) [closed]

So We need to show that our topology is power set of $$X$$. how can I proceed?

## closed as off-topic by Arnaud Mortier, Shailesh, Feng Shao, The Count, 0XLRAug 18 at 0:19

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Arnaud Mortier, Shailesh, Feng Shao, The Count, 0XLR
If this question can be reworded to fit the rules in the help center, please edit the question.

• It's wrong in general. Otherwise there wouldn't be two distinct names for it. This question lacks context as such. – Arnaud Mortier Aug 17 at 20:04
• What is $X$? This is not true unless you have a specific $X$ in mind. – 0XLR Aug 17 at 20:05
• I don't know the exact details of the question, just remember my professor saying something like this. But i think the new edit gives sufficient information. – Cosmic Aug 17 at 20:08
• With the new edit, the result is now true, but it's also really elementary, and still lacks context. Please read this post first. – Arnaud Mortier Aug 17 at 20:14

The cofinite topology on $$X$$ consists of the empty set and all cofinite sets, meaning all sets with finite complement. Since every subset of $$X$$ is finite, so is their complement. Therefore, the cofinite topology is equal to the power set.
Given any set $$X$$, we can define a topology $$\tau$$ on it, called the cofinite topology, by declaring the empty set $$\phi$$ and all the cofinite (look here for the definition of cofinite subset) subsets of $$X$$ to be the open sets (why is this collection form a topology on $$X$$?).
Now if $$X$$ is finite then the complement of any subset of $$X$$ is open in $$\tau$$ (why?).