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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?

haroba
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    A metric space has no inherent algebraic structure. – Brian M. Scott Jul 23 '12 at 07:00
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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? – haroba Jul 23 '12 at 07:08
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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 07:10
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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 08:06

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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).

Arturo Magidin
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  • I agree with leaving so-called "vector spaces" out of it. They're definitely best viewed as algebraic structures, even if there's an inner product floating around. Even Hilbert spaces really aren't "spaces" imo. – goblin GONE Oct 22 '15 at 09:40
  • So basically, a handwavy way of summarizing is: a space is a set with added structure, whereas an algebra is a set with added operations? Also, pretty insightful to count vector spaces as algebraic structures! – David Cian Feb 12 '21 at 23:39
  • Why would you not place vector spaces in the class of “spaces”? – gen-ℤ ready to perish Aug 21 '21 at 13:55
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    @gen-ℤreadytoperish: I explained why. Vector spaces are really algebraic structures. They have far more in common with things like groups and rings than they do with things like topological spaces or differentiable manifolds. Just because it has "space" in the name doesn't make them one in this classification. – Arturo Magidin Aug 21 '21 at 15:32
  • @ArturoMagidin There’s really no need to be condescending like that. I get that vector spaces are algebraic structures, which I pointed out. You didn’t define a “space” at all; you just listed out examples of spaces. I don’t see why a vector space couldn’t maybe be both. It’s just a question—but excuse me for trying to learn. Let me know if there’s anything else I can do to be put down so you feel smarter. – gen-ℤ ready to perish Aug 21 '21 at 15:48
  • Well, I pointed it out and then deleted that sentence before posting, but the rest of my response still stands. – gen-ℤ ready to perish Aug 21 '21 at 15:57
  • @gen-ℤreadytoperish: It is not condescending to point out that the explanation is in the post you are replying to. That you take particular offense that I did not realize that you had written something that you deleted, on the other hand, is quite amusing. The very answer points out that if you put extra structure on a vector space then it acquires some of the trappings of what I've put in the "space" bucket, but that by itself it simply doesn't have them. You are free to disagree, but given that I put the reasons in the answer to begin with, asking me to reiterate them seems wasteful. – Arturo Magidin Aug 21 '21 at 16:23
  • @gen-ℤreadytoperish: There is nothing in a structure of an abstract vector space that distinguishes it from the purely algebraic structures, and there is nothing in the structure that gives it the flavor of a topological, geomtric, differential, or metric space. It is only when you add structure (inner products, metrics, topologies) that they acquire the latter. Given the total absence of characteristics that make vector spaces akin to the other spaces I list, and the complete adherence to the qualities that make them similar to algebraic structures, there is simply no reason to put them there – Arturo Magidin Aug 21 '21 at 16:27
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There's no precise definition; anyway, the way I look at it:

  • There exist many different types of spaces (e.g. sets, posets, graphs, digraphs, metric spaces, uniform spaces, topological spaces, manifolds, etc.) There's no precise definition, but anyway, spaces usually form a distributive category. Furthermore, if $U : \mathbf{C} \rightarrow \mathbf{Set}$ denotes the relevant forgetful functor, then this usually preserves coproducts, and it usually has a left-adjoint $F$ such that the inclusion $UFS \leftarrow S$ is an isomorphism for each set $S$.

    (For this reason, I wouldn't consider vector spaces to really be "spaces".)

  • We may consider spaces that are equipped with further algebraic structure, like a set equipped with the structure of a group, or a topological space equipped with the structure of a group, or a manifold equipped with the structure of a group. Etc. These can often be defined as finite-product preserving functors out of an appropriately chosen Lawvere theory. Categories of algebraic structures usually don't form a distributive category, despite that all relevant products and coproducts often exist, and the relevant forgetful functor to the relevant category of spaces usually doesn't preserve coproducts. But I think the biggest giveaway that we're not dealing with spaces is that if $U$ and $F$ are the relevant free and forgetful functors, there tend to exist spaces $S$ such that the inclusion $UFS \leftarrow S$ is not an isomorphism.

goblin GONE
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A mathematical structure is a set or sets associated with some mathematical object (s) like a binary operation,collection of its subsets etc which satisfy some axioms.the mathematical object (s) is called structure and the set is called ground or underling set.example topological structure (X, tau) here tau is structure ans x is underlying set... similarly algebraic structure (x,*) now a space is a mathematical structure where the structure is of geometric type .for example (x,tau) here tau is a geometric type structurer so it is a space called topological space.

shah faisal
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