I am addressing only the first part of your question (i.e., nothing with the structure of vector space; only topology and limits of sequences).
I will quote here part of Problems 1.7.18-1.7.20 from Engelking's General Topology. (It would be better if you could get the book. I believe it used to be here, but the links don't work now. Perhaps you'll find it in the Internet.)
L*-space is a pair $(X, \lambda)$, where X is
a set and $\lambda$ a function (called the limit operator) assigning to some sequences of points of X
an element of X (called the limit of the sequence) in such a way that the following conditions are satisfied:
(L1) If $x_i=x$ for $i = 1,2,\dots$, then $\lambda x_i = x$.
(L2) If $\lambda x_i = x$, then $\lambda x' = x$ for every subsequence $x'$ of $x$.
(L3) If a sequence $\{x_i\}$ does not converge to $x$, then it contains a subsequence $\{x_{k_i}\}$ such
that no subsequence of $\{x_{k_i}\}$ converges to $x$.
These properties are sufficient to define a closure operator on $X$ (not necessary idempotent).
If $(X,\lambda)$ fulfills and additional condition
(L4) If $\lambda x_i = x$ and $\lambda x^i_j = x_i$ for $i = 1,2,\dots$, then there exist sequences of positive integers
$i_1, i_2,\dots$ and $j_1, j_2, \dots$ such that $\lambda x_{j_k}^{i_k} = x$.
L*-space $X$ satisfying (L4) is called an S*-space. The closure operator given by S*-space is idempotent.
Using this closure operator we get a topology, such that the convergence of the sequences is given by $\lambda$. A topology can be obtained from a L*-space (S*-space) if and only if the original space is sequential (Frechet-Urysohn).
References given in Engelking's book are Frechet [1906] and [1918], Urysohn [1926a], Kisynski [i960].
Frechet [1906] Sur quelques points du calcul fonctionnel, Rend, del Circ. Mat. di Palermo 22 (1906), 1-74.
Frechet [1918] Sur la notion de voisinage dans les ensembles abstraits, Bull. Sci. Math. 42 (1918), 138-156.
Kisynski [1960] Convergence du type L, Coll. Math. 7 (1960), 205-211.
Urysohn [1926a] Sur les classes (L) de M. Frechet, Enseign. Math. 25 (1926), 77-83.
NOTE: Some axioms for convergence of sequences are studied in the paper:
Mikusinski, P., Axiomatic theory of convergence (Polish), Uniw. Slaski w Katowicach Prace Nauk.-Prace Mat. No. 12 (1982), 13-21. I do not have the original paper, only a paper which cites this one; it seems that the axioms are equivalent to (L1)-(L3) and the uniqueness of limit. (But I do not know, whether some further axioms are studied in this paper.)
EDIT: In Engelking's book (and frequently in general topology) the term Frechet space is used in this sense, not this one. I've edited Frechet to Frechet-Urysohn above, to avoid the confusion.
$C_0(\Omega)$ is the closure of $C_c(\Omega)$ only with respect to the sup topology. The inductive limit topology on $C_c(\Omega)$ gives you a complete topological vector space.
The dual of $(C_0(\Omega),\tau_{sup})$ is the space of bounded (also called finite) Radon measures. The dual of $(C_c(\Omega),\tau_{ind})$ is the space of locally bounded Radon measures. At least, if $\Omega$ is $\sigma$-locally compact. The two spaces coincide if $\Omega$ is compact.
The space $(C_0(\Omega),\tau_{ind})$ does not make sense, because the inductive limit topology depends on the sequence of spaces you use to cover the entire functional space and, in general, you cannot cover $C_0(\Omega)$ through $C_c(U)$ spaces with $U$ included in $\Omega$.
Finally, the space $(C_c(\Omega),\tau_{sup})$ makes sense but bounded linear functionals on this space are naturally identified to bounded linear functionals on $(C_0(\Omega),\tau_{sup})$.
If you want to go deep into these questions you have to take a look to these references:
- Irene Fonseca, Giovanni Leoni. Modern Methods in the Calculus of Variations: $L^p$ Spaces. Springer Science & Business Media, 2007.
- Charalambos D. Aliprantis, Kim C. Border. Infinite Dimensional Analysis: A Hitchhiker's Guide. Springer-Verlag Berlin and Heidelberg GmbH & Company KG, 2013.
- Nelson Dunford, Jacob T. Schwartz. Linear operators, part 1: general theory. Vol. 10. John Wiley & Sons, 1988.
Best Answer
A topology is defined by somehow specifying which sets are open. One way to do this is to specify which sets are closed instead, as a set is open if and only if its complement is closed.
Suppose you have some concept of convergence of sequences in your space. Presumably, this concept has the property that any subsequence of a convergent sequence is also convergent, and to the same limit. Otherwise, you don't have an adequate concept of convergence.
You can define a topology on your space by saying a set $C$ is closed if and only if it contains the limits of all convergent sequences contained in it. Clearly the empty set and entire space are closed under this definition. If $C$ and $D$ are closed, then any convergent sequence in $C \cup D$ must have an infinite subsequence in one of them. That subsequence converges, and its limit must be in the same space. But that is also the limit of the original sequence, which therefore must be within $C \cup D$. Thus $C\cup D$ is closed. And finally any intersection of closed sets is closed, because any convergent sequences in the intersection are also in each of the closed sets, therefore so is its limit.
So this does define a topology. But there is a caveat. From a topology, you can define the concept of convergence of sequences. But that concept may not match the original concept you had for convergence of sequences. All your original convergent sequences will continue to converge to the same limits under this topology. But it is possible they may converge to other limits as well (i.e., the topology need not be T1). And other sequences that were not convergent in your original concept may converge in the topological sense.
This is why convergence of sequences is not a preferred method for defining topologies.