# Fréchet L-Spaces

NOTE: The question has now been posted on MathOverflow: 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 Googling.
Where could I find about these spaces?
Also I wonder; why this concept is not popular in mathematics?

• Look for convergence spaces. These are a generalisation of topological spaces in a similar way. The difference is that filters and their convergence is used for definition instead of the convergence of sequences. – Paul K Feb 18 '16 at 10:24
• Have you tried looking in Kelley's book? archive.org/details/GeneralTopology – Giuseppe Negro Feb 18 '16 at 10:27
• See: last chapter of Set Topology by Vaidyanathaswamy; History of Sequential Convergence Spaces in Handbook of the History of General Topology, Volume I; Angus Ellis Taylor's lengthy (217 pages in three parts) historical analysis of Frechet's work (see my answer here); Richard Friederich Arens, Note on convergence in topology, Mathematics Magazine 23 #5 (May-June 1950), 229-234. – Dave L. Renfro Feb 18 '16 at 15:51
• If anyone is interested, I looked through my copy of Kelley's book this morning and all I found was net and filter convergence -- nothing about Fréchet's $L$-spaces (in fact, nothing about notions that are closely related to $L$-spaces by Fréchet or by other people). – Dave L. Renfro Feb 22 '16 at 15:15
• You might be interested in Problems 1.7.18-1.7.20 in Engelking's book and references listed there. You can look also at my answer here. And of course, it might be reasonable to try to have a look at papers which cite the original Frechet's paper. – Martin Sleziak Feb 11 '17 at 10:52