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I need to determine whether the collection $\tau_{\infty} = \{ U | X-U \, \text{is infinite or empty or the whole set}\, X \}$ is a topology.

Obviously, if $X$ itself is finite, then, all we would have in $\tau_{\infty}$ is $\{ \{ X\}, \{\emptyset \} \}$, which would make $\tau_{\infty}$ the trivial topology (and mentioning the part about $X-U$ being infinite is pointless).

Now, let's consider the case where $X$ is either countably infinite or uncountably infinite. If $A$ is either countably infinite or uncountably infinite, $card(A) = card(A - \{x\})$, where $x \in A$, right? So, $U = \{x\}$ where $\{x\}$ is a singleton element of $X$ must be in $\tau_{\infty}$.

Now, I'm trying to show that it can't be a topology by showing that $X - (X - \{x\})$ is not infinite, but I don't have the nice property like I do in sigma algebras that it needs to be closed under complementation. All I have to work with here are intersections and unions. And everything I try isn't working. It could be possible that I'm wrong and it is a topology, but something is telling me that it's not.

So, I was wondering if somebody could give me some hints or steer me in the right direction. Just be ready to answer follow-up questions and be patient!

Thank you.

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    Here's an easier approach: construct two sets in $\tau_\infty$ whose intersection is finite. – 2017-02-13
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    @Omnomnomnom wouldn't $X$ and $\{ x \}$ be an example of that? – 2017-02-13
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    Sorry, I meant two sets in $\tau_\infty$ whose union is not in $\tau_\infty$ – 2017-02-13

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Hint: You know that $\{x\}$ is open for every $x\in X$. What sets can you form by taking unions of sets of the form $\{x\}$, for different values of $x$?

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    like countably infinite unions? Or even any kind of infinite unions? – 2017-02-13
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    I'm sorry but your hint really isn't helping... Can you say a little bit more? – 2017-02-13
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    Recall that the union (any kind of union) of open sets must be open. So if $\{x\}$ is open for each $x\in X$ then... – 2017-02-13
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    @OwenSizemore The arbitrary union of the sets $\{ x \}$ must be open? How does that help? – 2017-02-13
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    @JessyunBourne: What sets can you make as unions of things of the form $\{x\}$? – 2017-02-13
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    @OwenSizemore pretty much everything in $X$. – 2017-02-13
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    @OwenSizemore is this the discrete topology? – 2017-02-13
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    @JessyunBourne: Yep! – 2017-02-13
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    @OwenSizemore intuitively I'm kind of getting that, but how do I show it here? – 2017-02-13
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    @OwenSizemore because for any element of $X$, its complement is also contained in $X$? – 2017-02-13
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    Let us [continue this discussion in chat](http://chat.stackexchange.com/rooms/53572/discussion-between-owen-sizemore-and-jessyunbourne). – 2017-02-13