# Fréchet L-Spaces

According to the paper The emergence of open sets, closed sets, and limit points in analysis and topology famous mathematician Maurice Fréchet 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:

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 $$X$$.

If $$(x_n)\in S$$, then $$F(x_n)$$ was said to be the “limit of the sequence $$(x_n)$$" satisfying following two axioms:

$$A_1$$: If $$(x_n)$$ is a constant sequence whose value is $$a$$, then $$F(x_n)=a$$.

$$A_2$$: If $$F(x_n)=a$$, then for any sub-sequence of $$(x_n)$$ given by $$(x_{n_k})$$ we have $$F(x_{n_k})=a$$.

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?

• Two months ago, I ask the same question on Mathematics and no one answered. Later I thought that this would be more suitable to MathOverflow. – Bumblebee Apr 26 '16 at 7:42
• It seems that Terry Pratchett also studied $L$-spaces ;) – Loïc Teyssier Apr 26 '16 at 7:48
• Today, a Frechet L-space means a space with two properties: Frechet (also known as Frechet-Urysohn) and L. Thus, the title should better be "Frechet's L-space". BTW, (nowadays) a space is Frechet-Urysohn if every point in the closure of a set is a limit of a sequence, and a space is L if it is hereditarily Lindelof and nonseparable. These are not entirely unrelated notions. – Boaz Tsaban Apr 26 '16 at 8:44
• As a search strategy I tried Fréchet "Limit Spaces". – Chip Eastham Apr 27 '16 at 12:51
• Nowadays, an L-space is a regular hereditarily Lindelöf space that is not separable. I believe it's also a notion in metric topological vector spaces. A somewhat overloaded word. – Henno Brandsma Apr 30 '16 at 21:27

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

where these structures are named convergence spaces, more precisely sequential convergence spaces and are studied in chapter 1.7.

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

but I haven't read that one.