An ultrafilter product topology

Question

Suppose $X=\prod _{i\in\omega}X_i$ is the cartesian product of topological spaces $X_i$ and $u$ is a filter on $\omega$.

Define a basis for $X$ by taking the collection of all sets of the form $\prod _{i \in\omega} U_i$ where $U_i\subseteq X_i$ is open and $\{i\in\omega :U_i=X_i\}\in u$. It is easily checked that this is a basis.

This topology would coincide with the standard product topology ig $u$ is the filter of cofinite subsets of $\omega$. If we extend this to a free ultrafilter, then I think you have a topology properly between the product and box topologies. If $u$ is principal, I believe the topology would be close to the box topology.

I will be interested in the case that $u$ is a free ultrafilter.

Questions:

1) Are there any immediate properties/theorems concerning this space? Hopefully it can be interesting.

2) References?

Answer 1

These are some properties I could get without too much effort. Below $u$ is a free ultrafilter on $\omega$.

• The $u$-product of Hausdorff spaces is Hausdorff (since this topology is finer than the usual product topology).

• If the set $A = \{ n \in \omega : | X_n | = 1 \}$ belongs to $u$, then the $u$-product $\prod_{n}^u X_n$ is homeomorphic to the box product ${\large\Box}_{n \notin A} X_n$. If, additionally, $A$ is co-infinite, we may then show that certain topological properties are not preserved by appealing to box products.

  • Taking $X_n$ to be the two-point discrete space for all $n \notin A$ shows that the following properties are not necessarily preserved: second-countability, separability, compactness, countable compactness, sequential compactnes, σ-compactness, Lindelöfness.
  • Taking $X_n$ to be the real line for all $n \notin A$ shows that the following additional properties are not necessarily preserved: first-countability, connectedness, path-connectedness, metrizability.
  • Taking $X_n$ to be the Baire space for all $n \notin A$ shows that the following additional properties are not necessarily preserved: normality, hereditary (or complete) normality, perfect normality, paracompactness.

• Regularity appears to be preserved, and the usual proof that the product of regular spaces is regular demonstrates this:

  If $\mathbf{x} = \langle x_n \rangle_n \in \prod_{n \in \omega}^u X_n$ and $U \subseteq \prod_{n \in \omega}^u X_n$ is an open neighbourhood of $\mathbf{x}$, without loss of generality we may assume that $U$ is a basic open set: $U = \prod_n U_n$ where $U_n$ is open in $X_n$ and $A := \{ n : U_n = X_n \} \in u$. For $n \notin A$, as $x_n \in U_n$ by regularity there is an open $V_n$ such that $x_n \in V_n \subseteq \overline{V_n} \subseteq U_n$. Setting $V_n := X_n$ for each $n \in A$, it follows that $V := \prod_n V_n$ is an open neighbourhood of $\mathbf{x}$. Furthermore, $\overline{V} \subseteq \prod_n \overline{V_n} \subseteq U$.

(I am unaware of any references for such spaces.)