I was trying to examine examples of topologies as a self learning project (you can see the list here), and these questions arose:

  1. I couldn't spot a product topology where any set other than basis element is open in it - that is, a product topology where all open sets are not basis elements of the form U x V, where U and V are open in their respective spaces.
  2. Related to above: Can product topology on two finite topologies ever be infinite?
  3. Product of countably infinite sets is uncountable. Can I safely assume that product topology on a countable collection of spaces with countably infinite topologies is uncountable?
  4. Related to above: What can we say about countability of product topology on product of countable collection of finite sets (and hence topologies on each space must be finite)? Am I right in guessing it must be countable?
  1. The open unit circle in $\mathbb{R}^2$ is an example, or $\{(x,y): x \neq y\}$. The plane has the product topology wrt the usual topology on $\mathbb{R}$. These sets are open but not Cartesian products of two sets.

  2. The product of two finite topologies is always finite: it has a finite base, and we can only form finitely many unions from it.

  3. It surely is uncountable. Prove it. The standard base is countable, but in most cases there will be uncountably many different unions formed from it.

  4. It can and often will be uncountable, as in the case of the Cantor cube $\{0,1\}^\mathbb{N}$. e.g.

  • $\begingroup$ Thank you, for the answer and the pointers towards few questions I didn't ask! Out of context; has any good results been (or can be) evolved out of observing countability of topologies? With my limited knowledge, it looks pointless, just something to understand topologies better. $\endgroup$ – Jesse P Francis Oct 31 '18 at 2:38
  • $\begingroup$ @JessePFrancis Not countability of the topology (which is rare ) but descriptions of the topology by countable means is a recurring theme. A space can be first countable , second countable, have countable density, countable ccc, countable tightness etc. These are more advanced topics though (cardinal invariants). $\endgroup$ – Henno Brandsma Oct 31 '18 at 5:07

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