# When the box means nothing

When assuming the axiom of choice the product topology and box the topology are quite different when considering infinite products. For example the Tychonoff product of countably many sets of three elements is compact, metrizable an all in all a very nice space. On the other hand, the box product is not separable or second countable at all.

But without the axiom of choice the world is indeed a strange place. This was posted as answer on math.SE earlier today.

Definition. We say that $S$ is a Russell set if there exists a partition of $S$ into $P_n$ for $n\in\omega$, such that $|P_n|=2$ for all $n$, and for every infinite $I\subseteq\omega$, the product $\prod_{i\in I}P_i$ is empty.

Russell sets are infinite Dedekind-finite sets. They cannot be linearly ordered, and therefore witness both the failure of the axiom of countable choice (even for families of pairs, which is the least you can ask!) and the Boolean Prime Ideal theorem (which implies every set can be linearly ordered).

The existence of Russell sets is consistent, for example in Cohen’s second model there are Russell sets. Work in such model where a Russell set exists. Let $S$ be a Russell set, and $\{P_n\mid n\in\omega\}$ its partition into pairs witnessing this. We may assume that $S\cap\omega=\varnothing$, now define $X_n=P_n\cup\{n\}$ and $T_n=\{\varnothing,P_n,X_n\}$ is a topology on $X_n$.

Note that $\prod X_n$ is non-empty, but every choice function must pick co-finitely many ordinals. Therefore the space $\prod X_n$ is non-empty and so we can meaningfully define a topology on that space. Now look at the box topology defined there. If $U_n$ is a sequence of open sets, then $\prod U_n$ is non-empty if and only if $U_n\neq\varnothing$ for all $n$, and cofinitely often $U_n=X_n$.

This means that the box product is exactly the Tychonoff product!

Weird, huh?

The account has been suspended, I’d like to thank everyone who helped! I have removed the comments posted by “Isa Bria” after the real Isa Bria has contacted me and asked to remove them.

We have verified, in the meantime, that the same person impersonating me on Quora is the one who used Isa’s name in those comments.

Someone has been impersonating me on Quora. They use my real name, they link to my homepage and claim to be a set theory Ph.D. student from Jerusalem.

But it’s not me. Even if there was another Asaf Karagila out there in the world, I think it’s pretty thin that I wouldn’t have known him since we both study at the same place and the same topic. The homepage links HERE! This one!

Somehow the Quora management is convinced that the identity is real (I have reported it, but the “unverified” status was removed after a few hours).

The account is asking a lot about Iran and Nazism related questions, and politics related nonsense topics. Anyone who has ever met me knows how much I despise these topics, and those who know me will instantly recognize this is not my work. Here are a couple of screenshots, courtesy of a friend.

The last question in the second one is actually shows that this cannot possibly be me. Why would anyone who was born and raised in Israel ask where to go to vacation in Israel? Why would I need Quora to find recommendation about beaches, or hotels or cities?

Please report that account, who goes by my name. Please report every post they make, and please help Quora understand that this is not me. Please spread the word, ask your friends who have Quora accounts to join this effort, and help me restore my identity to its original owners.

If you have any additional thoughts, I’d be happy to hear them.

# A problem and a possible solution

So closing in on my third year, and in theory I should finish my dissertation by next summer. This means that I should probably start the writing process around April (I’m a fast writer, what with having a quality keyboard and knowing LaTeX quite well).

But if I want to be sure that I can finish next year, I should probably omit one of the problems I originally wanted to solve; and keep that for later, unless it turns out to be particularly simple when I finish the rest.

What do you think? Is it a good solution?