Let $(X,\tau)$ be a Hausdorff space. Denote by $\tau\_\text{seq}$ the topology on $X$ whose closed sets are the sequentially $\tau$-closed subsets of $X$. I have read that $\tau\_\text{seq}$ has the following properties:


1. $\tau\_\text{seq}$ is the strongest topology on $X$ for which the converging sequences are the $\tau$-converging sequences.
2. $f:X\to\Bbb R$ is sequentially $\tau$-lsc $\iff$ $f$ is $\tau\_\text{seq}$-lsc. (Here lsc means lower semicontinuous.)
3. $\tau\_\text{seq}=\tau$ if $\tau$ is a first-countable topology.


I would love to know more about properties of $\tau\_\text{seq}$ but the reference for the above claims is the article *Topologie e strutture di convergenza* by Dolcher and, unfortunately, I don't read or speak Italian.



> 
> Does anyone know a book/paper/article written in **English** that discusses the properties of $\tau\_\text{seq}$? For example, whether it is first-countable, completely regular, or locally convex, provided that we know what $\tau $ is.
> 
> 
> 


The particular case that is of interest to me is when $X$ is an infinite dimensional Banach space and $\tau = w$, the weak topology. $\tau\_\text{seq}$ is very related to the sequential $w$-lsc relaxation of a (nonlinear) functional $J:X\to \Bbb R$. This kind of relaxation is, in general, different from the topological $w$-lsc relaxation and is more useful in optimization and calculus of variations.



Properties of the topology of sequential convergence $\tau_\text{seq}$
Concerning the sequential coreflexion $w\_{seq}$ of the weak topology on a Banach space $X$ the following characterization can be proved.



> 
> **Theorem.** For a Banach space $X$ the following conditions are equivalent:
> 
> 
> 1) $X$ is reflexive;
> 
> 
> 2) $(X,w\_{seq})$ is a locally convex topological vector space;
> 
> 
> 3) the addition operation $+:X\times X\to X$ is jointly continuous with respect to the topology $w\_{seq}$.
> 
> 
> 


*Proof.* (1)$\Rightarrow$(2) If $X$ is reflexive, then the closed unit ball $B$ of $X$ is compact in the weak topology. Moreover, it is Eberlein compact and hence Frechet-Urysohn, which implies that on each ball $n\cdot B$ the topology $w\_{seq}$ induces the weak topology. Since each weakly convergent sequence is bounded, the topology $w\_{seq}$ coincides with the topology of the direct limit $\varinjlim nB$ of the sequence $(n B)\_{n\in\mathbb N}$, which implies that $(X,w\_{seq})$ is a $k\_\omega$-space. Now the continuity of the addition $+:nB\times nB\to 2nB$ in the weak topology implies the continuity of the addition in the topology $w\_{seq}$ of direct limit $\varinjlim n\cdot B=(X,w\_{seq})$. By the same reason, the multiplication map $X\times\mathbb R\to X$ is continuous with respect to the topology $w\_{seq}$. So, $(X,w\_{seq})$ is a linear topological space. Its local convexity can be proved using the local convexity of the weak topology and the coincidence of the topology $w\_{seq}$ with the direct limit topology $\varinjlim nB$ of the sequence of compact convex sets.


(2)$\Rightarrow$(3) is trivial.


(3)$\Rightarrow$(1) Assumining that $X$ is not reflexive, we conclude that the closed unit ball endowed with the weak topology is not compact and hence not sequentially compact (by the classical [Eberlian-Smulian Theorem](https://en.wikipedia.org/wiki/Eberlein%E2%80%93%C5%A0mulian_theorem)). Consequently, $X$ contains a non-reflexive separable Banach subspace $Y$. Assuming that the addition operation is continuous with respect to the topology $w\_{seq}$ on $X$, we conclude that it is continuous with respect to the topology $w\_{seq}$ on $Y$ and $(Y,w\_{seq})$ is a topological group. 


The separability of $Y$ implies that the closed ball $nB\_Y$ of radius $n$ is metrizable and separable in the topology $w\_{seq}$ which coincides with the weak topology on $nB$. Consequently, the metrizable separable space $(nB,w\_{seq})$ has countable base $\mathcal B\_n$ of the (weak) topology.
The union $\mathcal B=\bigcup\_{n\in\mathbb N}\mathcal B\_n$ is a *countable $cs$-network* at zero of the space $(Y,w\_{seq})$. The latter means that for any sequence $\{x\_n\}\_{n\in\omega}\subset X$ that converges to zero in the topology $w\_{seq}$ and any neighborhood $U\in w\_{seq}$ of zero there exists a set $B\in\mathcal B\_n$ such that $0\in B\subset U$ and $B$ contains all but finitely many points $x\_n$.


By a result of [Banakh and Zdomskyy](https://arxiv.org/abs/0911.4075), a sequential topological group having a countable $cs$-network at zero is either metrizable of contains an open $k\_\omega$-subgroup. But $(Y,w\_{seq})$ is neighter metrizable nor contains an open $k\_\omega$-subgroup. This contradiction shows that $(Y,w\_{seq})$ is not a topological group and the addition is discontinuous. 




---


**Remark.** Some topologies near to $w\_{seq}$ have been considered in the paper [[T.Banakh, *On topological classification of normed spaces endowed with the weak topology or the topology of compact convergence*]](https://arxiv.org/abs/1908.09115), published in [this book](https://rads.stackoverflow.com/amzn/click/com/1560729783).


