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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?

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The paper is not accessible to me (paywall), but from the first page, which is shown to me, it seems that such construction is at least mentioned in C. J. Knight: Box Topologies dx.doi.org/10.1093/qmath/15.1.41 In this survey it is called ideal product topology. –  Martin Sleziak Jul 16 at 9:22

1 Answer 1

up vote 4 down vote accepted

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.)

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