Talk:Sequential space

History

Is the history about Franklin correct? The book I'm reading (Archangelskii and Pontryagin, General topology I) goes on about sequential spaces and their properties without mentioning Franklin; and tacks on the names of Frechet and Urysohn to various defintions. Frechet and Urysohn are a couple of generations older than Franklin in 1965 ... It is implied that Franklin gave a category theory definition, is that right? That would make more sense to me. linas 16:02, 6 November 2006 (UTC)

Lots of textbooks go on about definitions of things without mentioning their first occurence in the literature. It's been a while since I read Franklin's original 1965 paper, but I seriously doubt Frechet and Urysohn defined "sequential space", as it is the class of sequential spaces which are exactly specified by the question that Franklin was trying to answer in his 1965 paper. And the 1965 paper did have category theory in it, if I remember, but the basic definition were just in terms of top. spaces. But, you should really read the original paper to be sure. Revoler

Quote from Ryszard Engelking's book General topology (which has nice historical and bibliographic notes at the end of each section): Sequential and Frechet spaces belonged to the folklore almost since the origin of general topology, but they were first thouroughly examined by Franklin in [1965] and [1967]. (This is pretty much the same as we have in the article now and I find historical notes in Engelking's book a reliable source.) --Kompik 09:12, 15 October 2007 (UTC)

Interwiki

I do not speak Polish, but have some basic understanding, since it is similar to Slovak language. I do not think that interwiki to pl:Przestrzeń Frécheta (topologia) is correct. As far as I understand the definition given in the Polish article, 'Przestrzeń Frécheta' means Frechet-Urysohn space. I do not know what is the Polish term for the sequential space. Someone with better knowledge of Polish language could perhaps correct this.

A related question: In English wiki there is no separate article for Frechet-Urysohn spaces (they are defined here, in the article on sequential spaces). Polish wiki obviously has one. Are there some rules how to use interwiki in such asymmetric cases? --Kompik 07:40, 30 June 2007 (UTC)

Cartesian closed

I've been led by multiple sources to believe that the subcategory of sequential spaces in Top is cartesian closed. For example, Booth and Tillotson prove that it is the smallest "convenient" topological category, which in particular means it is cartesian closed with exponential equipped with the (convergent sequence)-open topology.

P.I. Booth, A. Tillotson, "Monoidal closed, cartesian closed and convenient categories of topological spaces" Pacific J. Math. , 88 (1980) pp. 35–53.

See also this abstract: [1]

The categorical properties section of this article seems contradictory to this. What is the problem? - 129.100.75.90 20:59, 25 August 2007 (UTC)

BTW a preprint of the paper you mentioned can be found at [2]. There are many other papers which worth reading if you are interested in this topic. I mention also this one: Cartesian closed coreflective subcategories of the category of topological spaces, J Cincura, Topology Appl, 1991. In case you do not have access to the journal Topology and its Applications, let me know at my talk page -- I can send you a scanned copy of this paper, in case you think it could be interesting for you. --Kompik 09:17, 26 August 2007 (UTC)

What precisely do you find contradictory? A cartesian closed subcategory of Top need not be closed under topological products. The product in the category of sequential spaces is the sequential coreflection of the usual product. The same holds for products and limits in coreflective subcategories in general. {The sequential coreflection is the topology obtained by taking as closed sets the sequentially closed sets of the original topology. By sequentially closed I mean closed under limits of sequences. In the other words, a sets is sequentially closed iff the sequential closure of A us A - the sequential closure is defined in the article.) BTW I think it's worth mentioning that this category is cartesian closed. --Kompik 09:11, 26 August 2007 (UTC)

Of course I figured this out last night after further reflection, but I wasn't able to get online to say so. Thanks for all the great references, though. I will make a few changes to the article to reflect this information. - 129.100.75.90 17:16, 26 August 2007 (UTC)

I've modified this part of the article:

The subcategory Seq is a cartesian closed category with respect to its own product (not that of Top). The exponential objects are equipped with the (convergent sequence)-open topology. P.I. Booth and A. Tillotson have shown that Seq is the smallest cartesian closed subcategory of Top containing the underlying topological spaces of all metric spaces, CW-complexes, and differentiable manifolds and that is closed under limits, colimits, subspaces, quotients, and other "certain reasonable identities" that Norman Steenrod described as "convenient".

Sequential spaces are definitely not closed under subspaces. It is possible that the article of Booth and Tillotson says something similar to hits, but the above was definitely not true. --Kompik (talk) 12:51, 28 April 2010 (UTC)

Categorical properties

I'm facing two contradictory results...

1. Here a "proof" that ${\mathcal {D}}(\Omega )$ (smooth functions with compact support endowed with the inductive limit topology, category of locally convex top. vector spaces) is not sequential. (${\mathcal {D}}(\Omega )$ is the inductive limit of $({\mathcal {D}}_{K_{n}})_{n\in \mathbb {N} }$, smooth functions with support in the compact $K_{n}$ (exhaustion of $\Omega$ by compact sets). These are normed spaces, so they are sequential)

1. in the article, the statement that the inductive limit of sequential spaces is sequential

Noix07 (talk) 17:57, 17 March 2019 (UTC)

It is probable that the inductive limit in the category of locally convex topological spaces is not the inductive limit in the category of sequential spaces.Noix07 (talk) 15:55, 19 March 2019 (UTC)

Sequentially Hausdorff

On the current page, there are 2 definitions of sequentially Hausdorff which are contradictory:

The first definition is at section "Definition"-"Preliminary", where it says "a space in which every sequence converges to at most one point is called a sequentially Hausdorff space"

The second definition is in the section "Topology of sequentially open sets" where it says "The topological space $(X,\tau )$ is said to be sequentially Hausdorff if $\operatorname {SeqOpen} (X,\tau )$ is a Hausdorff space."

According to the answer to question #here, the first definition is strictly weaker than the second, with a counterexample that is sequential space and every convergent sequence has a unique limit, but not Hausdorff. He suggests the first should be called as US(Unique Sequential Limit) instead. --MCXZX (talk) 11:01, 21 February 2022 (UTC)

Bernanke's Crossbow, Mgkrupa please take a look. There are indeed two conflicting meanings for sequentially Hausdorff.

A google search seems to show that the meaning of "unique sequential limits" is more common. I think the name "US-space" is a better choice, as there is no ambiguity there. However I would suggest to delete the sentence "A space in which convergent sequences have unique limits is called sequentially Hausdorff after the analogous condition for nets; every Hausdorff space is sequentially Hausdorff." altogether from the Preliminary definitions section, as it does not add anything to the discussion of sequential spaces.

As for the definition of sequentially Hausdorff in terms of the sequential coreflection of the space, the only ref I was able to find with this meaning is https://www.researchgate.net/publication/332530524_Sequentially_Hausdorff_and_full_sequentially_Hausdorff_spaces. Not sure who added that def in the article, but do you have any other references with this meaning? (Not even sure we need to keep this definition here, but I am curious to see if this is used somewhere) PatrickR2 (talk) 00:14, 2 July 2022 (UTC)

Additional: the notation $\lim x_{\bullet }$ to indicate the only limit point of a sequence is not used anywhere in the article. So this can also be removed from the article. Let's keep the clutter down! PatrickR2 (talk) 00:30, 2 July 2022 (UTC)

I can't remember seeing that definition in terms of sequential coreflection. Since "sequentially Hausdorff" is not used in the article, I've removed both definitions (if you wanted them changed instead of removed then feel free to revert my edit). Also feel free to remove $\lim x_{\bullet }.$ I'll leave the choice of notation up to you. Mgkrupa 05:52, 2 July 2022 (UTC)

Thanks for removing both definitions. I think that's the right thing for now. PatrickR2 (talk) 06:59, 2 July 2022 (UTC)