Compact vs. Linearly Compact

Question

Definition 1: A topological $R$-module $M$ is linearly compact in case every collection of closed linear varieties in $M$ with the finite intersection property has a nonvoid intersection (Daniel Zelinsky, Linearly Compact Modules and Rings). The phrase linear variety in $M$ to mean a coset of a submodule of $M$.

In topological group, we have every open subgroup is a closed subgroup.

Question 1: Can we restate Definition 1 as:

Definition 1': A topological $R$-module $M$ is linearly compact in case every collection of open linear varieties in $M$ with the finite intersection property has a nonvoid intersection?

Question 2: Why does it use closed linear varieties in a module like an open cover of $M$ in the definition of compact space? How do closed linear varieties in $M$ with the finite intersection property cover $M$?

Question 3: What is the motivation for a linearly compact module? Why, in topological algebra, are many authors interested in linearly compact space rather than compact space for topological modules?

Thank you in advance for considering my questions.

Answer 1

I am not an expert in the area, but let me try to give you some answers.

1. Your definition is not equivalent to the original definition. First of all, even though open subgroups are closed, the converse is very rarely true. Take for example the trivial subgroup $\lbrace 0\rbrace$, which is only open in discrete subgroups. Further, linear varieties are very rarely subgroups anyway - a non-trivial coset of the subgroup $H<G$ would have the form $a+H$ for $a\in G\setminus H$, so in particular $0\notin a+H$. I haven't been able to cook up a concrete counter-example off the top of my head, but I imagine it is doable.

2. The intersection property is related, but not completely analogous, to the open cover property. In particular, the linear varieties shouldn't constitute an open cover of the module. See 3) for how they are related.

3. Firstly, being linearly compact is a weaker property than being compact, and thus more flexible. If you follow Bruno B's link, you will see that being compact is equivalent to having the finite intersection property for any collection of closed subspaces, so in particular also for linear varieties. The true reason for introducing this notion and the needed flexibility, rather than working with compact modules, is rather deep. See Tom Leinster's blog post https://golem.ph.utexas.edu/category/2012/09/where_do_linearly_compact_vect.html, as well as his question on stack-overflow https://mathoverflow.net/questions/104777/what-are-the-algebras-for-the-double-dualization-monad/104845#104845. In essence, it is about finding the types of object that parallels compact spaces in the module setting - and here translating it directly to compact modules becomes too restrictive.