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A mathematical space is defined as:

A set (sometimes called a universe) with some added structure.

Well, a mathematical group is:

A set defined with a combining operation and some additional properties.

A set with some added structure.

Does this count as extra "structure", and could you then consider a group a type of "space"? How about rings and fields, can they be considered types of spaces? What about other objects like sheaves? Basically can any object which is a set with extra stuff around the set be considered a space, or is there something that space is unique from these other objects? If groups/rings/fields/sheaves are not spaces, why not?

Lance
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    Isn't this essentially the same as this question? – Arturo Magidin Jan 08 '22 at 02:36
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    What you quote are not definitions; they are informal descriptions. There is a formal definition of what a "group" is, but the general notion of "space" is not a formal notion, and there is no precise, formal definition. – Arturo Magidin Jan 08 '22 at 02:42
  • One can drive themselves mad over distinctions, so I prefer to think of intuitive appeal; certainly algebraic terms and the notion of a space overlap, but the reason why space was chosen for some objects is to highlight the natural geometry. A topology is an abstract set system, but a topological space introduces a context in which the topology lives. Analogously, a group acts on a space, but one would not say that group is a space. – While I Am Jan 08 '22 at 02:50
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    The short answer is that a "space" isn't generally defined. We define various kinds of spaces, e.g. topological spaces, metric spaces, measure spaces, Euclidean spaces, vector spaces, etc. "Space" is rather an idea or concept rather than a single unified thing. – AJY Jan 08 '22 at 03:09
  • Is a group a set with additional structure? – John Douma Jan 08 '22 at 03:11
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    It's not really worthwhile to fret too much about what some wikipedia writer meant, especially when it's one of those broad, intuitive, opening sentences of the wikipedia article. – Lee Mosher Jan 08 '22 at 03:14
  • By the author's definition the answer is yes. The language of groups has a binary function symbol and each group consists of a set and an interpretation of the binary function symbol, i.e. it defines multiplication. I really don't understand the other comments since the author defines what he means by a space. – John Douma Jan 08 '22 at 14:58

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