Is a mathematical group also a mathematical "space"? [duplicate]

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?

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).

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.