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Common algebraic categories are group, ring, module and algebra. Some of them have the corresponding object in topology, like topological group and topological linear space. We define them by making operations continuous or smooth. So there are two questions left:

(1) Is it reasonable to generalize topological linear space to topological module? And instead of $\mathbb R$ or $\mathbb C$ as coefficient in functional analysis, can we use a ring $R$?

(2) How about topological ring and topological algebra? Are there any reference about them?

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gaoxinge
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  • I imagine that the big barrier to using $R$ instead of a number field for functional analysis is that a lot of the current techniques rely on norms, which iirc are not easily generalizable to other contexts. – Eric Stucky Mar 21 '14 at 12:59
  • (1): Yes. You need a topological ring $R$ of course. The nicer $R$ is, the more you get from a topological module structure. (2): Yes. Bourbaki has some things in that respect. I don't know whether anything but topological vector spaces, Banach algebras and general topological groups is used much, but they are reasonable concepts, and have been investigated. – Daniel Fischer Mar 21 '14 at 13:01
  • @DanielFischer Topological vector spaces over the p-adics are a hot topic. See http://en.wikipedia.org/wiki/Berkovich_space, for example. – arsmath Mar 21 '14 at 13:16
  • @arsmath Yes. Those were included in my mention of topological vector spaces, but as the OP mentioned only $\mathbb{R}$ and $\mathbb{C}$ as scalar fields, it's good to make that explicit, thanks. – Daniel Fischer Mar 21 '14 at 13:22

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As for Topological Ring and Topological algebra are concerned they are important topics to study, for examples see this http://en.wikipedia.org/wiki/Topological_ring and http://en.wikipedia.org/wiki/Topological_algebra... infact there is a very nice book written on Rings of Continuous Functions by Gillman and Jerrison. The fact that study of topological groups find more applications is because they can be used to study symmetries and they first arose while studying groups of continuous transformations, both of which find immense utility in physics also.

wanderer
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These are all reasonably well-developed theories. Bourbaki's General Topology has chapters on general topological rings and modules. Warner has two monographs, Topological Rings and Topological Fields that go into more detail.

arsmath
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