According to the paper [The emergence of open sets, closed sets, and limit points
in analysis and topology](http://dx.doi.org/10.1016/j.hm.2008.01.001) famous mathematician [Maurice Fréchet](https://en.wikipedia.org/wiki/Maurice_Ren%C3%A9_Fr%C3%A9chet) who introduced the concept of metric spaces has also introduced another similar class of abstract spaces called **Limit spaces** based on the primitive idea of the limit of an infinite sequence in 1904, which was defined as follows:



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> An L-space is a set $X$ together with a function $F : S\to X,$ where $S$ is a set of infinite sequences of members of $X$.
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If $(x\_n)\in S$, then $F(x\_n)$ was said to be the *“limit of the sequence $(x\_n)$"* satisfying following two axioms:



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> $A\_1$: If $(x\_n)$ is a constant sequence whose value is $a$, then $F(x\_n)=a$.
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> $A\_2$: If $F(x\_n)=a$, then for any sub-sequence of $(x\_n)$ given by $(x\_{n\_k})$ we have $F(x\_{n\_k})=a$.
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I would like to know more about mathematics on L-spaces. But I could not find any thing by just Googling.  

Where could I find about these spaces?



Fréchet L-Spaces
I think that these spaces don't go under the name of $L$ spaces anymore. Actually, I am not sure if there is a consensus on how these structures are called today.


A good place to start is the fairly recent book


* [**Convergence Structures and Applications to Functional
Analysis**](https://books.google.com/books?id=OdwXRK-i9YgC&lpg=PP1&hl=de&pg=PR4#v=onepage&q&f=false) by R. Beattie and H.-P. Butzmann, Kluwer, 2002


where these structures are named **convergence spaces**, more precisely **sequential convergence spaces** and are studied in [chapter 1.7](https://books.google.com/books?id=OdwXRK-i9YgC&pg=PA51).


Another, slightly older, reference is the article (in French)


* [**Convergence du type $\mathcal{L}$**](https://eudml.org/doc/210660), Jan Kisyński, Colloquium
Mathematicae, 1960


but I haven't read that one.


