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What is the difference between a "space" and an "algebraic structure"? For example, metric spaces and vector spaces are both spaces and algebraic structures. Is a group a space? Is a manifold a space or an algebraic structure, both or neither?

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A metric space has no inherent algebraic structure. –  Brian M. Scott Jul 23 '12 at 7:00
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As I understand it an algebraic structure with at least one operation defined on it. Isn't a metric space an algebraic structure where that operation is "distance", with the regular axioms? –  Aqwis Jul 23 '12 at 7:08
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The metric isn’t in general an operation on the space $X$: it doesn’t take values in $X$. –  Brian M. Scott Jul 23 '12 at 7:10
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Guess that the terminology is not too canonical, but I'd say that an "algebraic structure" on a set $X$ is the datum of one or more possibly interacting operations on $X$. By a "space" $X$ we mean a set with some extra-structure of geometric type, such as a metric or a topology. –  Andrea Mori Jul 23 '12 at 8:06

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up vote 6 down vote accepted

If we take "algebraic structure" to be a synonym for "algebra" (in the sense of universal algebra), then an algebraic structure is a set $X$, together with a family of operations on $X$.

Recall that given a set $X$, an "operation" on $X$ is a function $X^{\alpha}\to X$, where $\alpha$ is an ordinal. Such a function is called an $\alpha$-ary operation; when $\alpha$ is a natural number, the operation is said to be "finitary" (takes only finitely many arguments).

Sometimes, algebraic structures are further enriched with (i) "partial operations" (functions defined on a subset $A\subseteq X^{\alpha}$ rather than all of $X^{\alpha}$), or (ii) $\beta$-ary relations (subsetes of $A^{\beta}$). We can also impose identities (requires that the operations/relations satisfy certain properties such as commutativity, etc).

In this sense, vector spaces, groups, rings, fields, etc. are all (enriched) "algebras"; metric spaces are not.

"Space" is a bit fuzzier; I would not put "vector spaces" in the class, restricting it rather to things like topological spaces, manifolds, metric spaces, normed spaces, etc.

Now, one should realize that you this does not have to be a dichotomy: you can have structures that include both kinds of data: a topological group is both an algebraic structure (a group) and a space (topological space), in a way that makes both structures interact with one another "nicely". Normed vector spaces are both algebraic structures (vector spaces), and "spaces" (normed spaces, hence metric, hence topological), where, again, we ask that the two structures interact nicely.

In fact, there is a lot of interesting stuff that can be obtained by having the two kinds of structures and "playing them off against one another." For example, Stone Duality and Priestley Duality exploit this kind of "structured topological space" (a topological space that also has operations, partial operations, and relations that interact well with the topology).

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