Not signed in (Sign In)

Start a new discussion

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Discussion Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJul 18th 2014
    • (edited Jul 18th 2014)

    I have touched net, just adding some more hyperlinks and cross-references within the page. Also filter, where I made eventuality filter come out as a link.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeApr 23rd 2017
    • (edited Apr 23rd 2017)

    I have been expanding at net a fair bit, also reformatting (more numbered environments) and re-arranging (more subsections, the order of some paragraphs reversed, where it seemed to help the reader).

    Mainly I wanted to, and did, add statement and detailed proof of these four facts:

    1. nets detect the topology

    2. nets detect continuity

    3. nets detect Hausdorff property

    4. nets detect compactness.

    (The last of these statements I also copied to a stand-alone entry compact spaces equivalently have converging subnet of every net. )

    Most of the other edits to the entry that I did were meant to make the statement of these proofs flow nicely. To that end for example I made explicit the classes of examples of directed sets that are needed in the proofs, in a Backgrounds-subsection “Directed sets”.

    And since I added all these proofs in the subsection “Relation to topology” (now “Properties – Relation to topology”) I moved the plain definition of convergence of nets in topological spaces, which used to be there, to the section Definition – Nets.

    At other places I simply expanded out the originally somewhat terse discussion to (hopefully) more easily readible text, such as in the section Nets and filters.

    Finally, I considerably expanded the Idea-section.

    All in all, while I did edit a lot, I tried to retain everything that used to be there, if maybe slightly re-arranged. But Toby should please check if he can live with the edits.

    • CommentRowNumber3.
    • CommentAuthormaxsnew
    • CommentTimeApr 24th 2017

    The correspondence betwen nets and filters seems very reminiscent of the correspondence between fibrations and presheaves, especially the construction of a filter net as a directed system of pointed sets seems very similar to the Grothendieck construction

    Is there any way to make this analogy more precise?

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)