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I'm trying to provee that the cofinite topology is a valid topological space. I've defined it as

$$C=\{\emptyset\}\cup\{S\subseteq X: X-S \quad\text{is finite }\}.$$

Now, clearly $\emptyset$ and $X$ are in $C$. I just have to prove that it is closed under infinite union and finite intersection:

1) We need to show that $$V=\bigcup_{i\in I}V_i\in C \quad\forall i\in I$$ where $I$ is the indexing set. Proving that $V$ is open is the same as showing that the complement is closed; now with the help of De Morgan's laws we have:

$$X-V = X-\left(\bigcup_{i\in I}V_i\right)=\bigcap_{i\in I}\left(X-V_i\right)$$

it's obvious that if one of the $V_i$, say $V_0$ is non-empty, we have

$$\bigcap_{i\in I}\left(X-V_i\right)\subseteq X-V_0$$ which is finite. Otherwise, every $V_i$ is empty and the union is empty therefore open

2) We need to show that $$V=\bigcap_{i=1}^nV_i\in C \quad $$ where $I$ is the indexing set. Proceeding in a similar manner as before: $$X-V = X-\left(\bigcap_{i=1}^nV_i\right)=\bigcup_{i=1}^n\left(X-V_i\right)$$

Which shows that we have finite unions of finite sets which will be again finite. Therefore, $\bigcap_{i=1}^nV_i\in C$

Is my logic correct? Have I missed something?

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Nope. Looks good. –  Sammy Black May 15 at 5:14

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