TITLE: Generalizing limits of sums, products, and quotients of sequences to abstract topological spaces?
QUESTION [13 upvotes]: Introductory real analysis books usually prove a list of properties about limits of sequences of real and complex numbers. Suppose $\lim x_n=x$ and $\lim y_n=y$. Then:

$\lim (x_n+y_n) = x+y$
$\lim c x_n = cx$ (where $c \in F$ for some field $F$)
$\lim x_n y_n = xy$
$\lim x_n^p=x^p$
$\lim \frac{1}{x_n} = \frac{1}{x}$ (provided for all $n$, $x_n \ne 0$ and $x \ne 0$)
$\lim \frac{x_n}{y_n}=\frac{x}{y}$ (provided for all $n$, $x_n \ne 0$, $y_n \ne 0$, $x \ne 0$ and $y \ne 0$)

However, these results can be generalized beyond sequences of real & complex numbers to more abstract sequences in more general metric and topological spaces. I would like to prove these results for an arbitrary metric space. However, the problem is, as far as I know, these operations (addition, multiplication and division) are only defined for numbers. So in an abstract metric or topological space, does $x_n+y_n$ have meaning?
How would one generalize these properties to abstract metric and topological spaces? Should you start with topological field extensions, topological algebras over a topological field, normed algebras, or something else? And just how general can you make these results? What is the most general topological space in which the operations $x_n+y_n$, $x_n \times y_n$, $\frac{x_n}{y_n}$, etc. make sense and all 6 properties of limits hold? Thanks for your thoughts!

REPLY [3 votes]: To talk about things like addition, multiplication, multiplicative identity, unlimited division, scalars from $F$ and for the notation $\frac{a}{b}$ to make sense, you're already talking about a field extension of $F$.
Then (almost by definition) a structure satisfies your list of properties $\iff$ it's a topological field which is also a field extension of $F$. There is some redundancy in the list:

$(3)\implies(2)$
$(3)\implies (4)_{p\ge0}$
$(3)\land(5)\implies(4)_{p\in\Bbb Z}$
$(4)_{p\in\Bbb Z}\implies(5)$
$(3)\land(5)\implies(6)$

Just throw away $(4)$ from the outset. We can get some other structures by discarding other conditions:

If one throws out $(2)$, then we have a topological field with no relation to $F$. If it is locally compact and not discrete, this is called a local field, and the topology is induced from a metric which is itself induced from an absolute value function.
If one throws out $(6)$ then multiplication isn't necessarily commutative so it is a topological division algebra, aka skew field, that contains the field $F$.
If one throws out $(2)\land(6)$, it's a topological skew field with no relation to $F$.
If one throws out $(5)\land(6)$ then it is a topological ring that contains $F$.
If one throws out $(2)\land(5)\land(6)$ then it is a topological ring.
If one throws out $(2)\land(3)\land(5)\land(6)$ then it's a topological abelian (additive) group.
If one throws out $(1)\land(2)\land(5)\land(6)$ then it's a topological (multiplicative) group.
If one throws out $(3)\land(5)\land(6)$ then it's a topological vector space over $F$.

Essentially, first figure out the algebraic structure you want, then add the word topological in front of it. One can still get things that don't have names, I suppose. (For instance, consider a topological ring in which inversion is continuous on the subspace of two-sided units, or take one of the above listed objects and append topological conditions on.)