Assume that $X$ is a topological space and $X^{\alpha}=X\times X\times\cdots$ is equipped with product topology. You may take $\alpha$ to be countable or uncountable. I want an "interesting" property which is not in $X$ but appears in $X^{\alpha}$.Usually, we see that properties of $X$ disappears in the product. I am curious about the opposite.
By "interesting", I meant properties not like cardinality of sets but like the "topological" properties mentioned in this post. I also don't want properties like "not P" (for example "not metrizable") where we already know $X$ has P but $X^\alpha$ doesn't.
Since the projection map transfers most of the properties like connectedness etc. on $X^{\alpha}$ to $X$,we need to look for properties which are not preserved by continuous, open surjective maps. Do you have any examples?
If you consider a property (not necessarily topological) to be interesting to you and worth noting then please mention them.
I know I have put too much restrictions on the type of properties I want and it is kind of vague. I'm sorry for that and I hope that you find those restrictions necessary. Please answer the question written in bold.
Thank you.
When I first thought of this, I remembered a meme where it says that since $1^{365}=1$ and $(1.01)^{365}=37.78$ so keep progressing everyday or something similar. I don't remember exactly what it said.
