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nLab > Latest Changes: extremally disconnected topological space

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeNov 21st 2013
    • (edited Nov 21st 2013)
    • CommentRowNumber2.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 29th 2019

    More redirects

    diff, v5, current

  3. I added a reformulation of being extremally disconnected as a left lifting property with respect to a proper surjective morphism of finite topological spaces, and some remarks about interpretation the Gleason theorem in terms of weak factorisation systems generated by proper morphisms of finite topological spaces.

    Anonymous

    diff, v7, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeOct 7th 2021

    Thanks.

    I have fixed your link (here).

    Also I tried to touch the wording, for clarity.

    But do you mean to say that

    the existence of a weak factorization system generated by this morphism implies that each space admits a

    ?

    I have trouble making sense of this. Don’t you just mean:

    “The right lifting property against these morphisms implies that …”?

    diff, v8, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeOct 7th 2021

    I have edited more:

    Merged the bit from the beginning of the entry where you introduced the lifting property with the bit you had later on deducing the Gleason theorem via lifting. Now both are in one subsection “Properties – As a lifting property” (here).

    Also adjusted the wording yet a bit more. Please do check if you can live with it. I am just trying to make it read more transparently (there is still room left to improve on that, I think). But if I broke something that you think shouldn’t be broken, please bear with me and fix it.

    diff, v8, current

    • CommentRowNumber6.
    • CommentAuthorGuest
    • CommentTimeOct 7th 2021
    Essentially, I meant the argument given at the page on separation axioms in terms of lifting properties,
    for which we do not have necessary notation at this page.

    >“The right lifting property against these morphisms implies that …”

    This is not enough: you need to _construct_ (find) some surjection onto $X$....

    In more detail, this means : existance of weak factorisation system (f_e.d.)^l (f_e.d.)^lr generated by f_e.d.
    implies that each map $\varnothing\to X$ decomposes as
    $$ \varnothing \xrightarrow{( f_{e.d.)^l} S \xrightarrow{ (f_{e.d.})^{lr}} X$
    where $S$ is extremally disconnected, and $S\to X$ is surjective and proper.

    With this, the right lifting property $ \varnothing \xrightarrow{( f_{e.d.)^l} ⧄ (proper) S implies/means
    that extremally disconnected spaces are projective.

    But I do not think we have the notation orterminology to give this argument here.
    • CommentRowNumber7.
    • CommentAuthorGuest
    • CommentTimeOct 7th 2021
    Re #5. It seems fine!

    One might want to give clearer explanation instead of

    >Both being surjective and being proper are right lifting properties. Hence, the existence of a weak factorization system generated by this morphism implies that each space admits a surjective proper map from an extremally disconnected space.

    what is said in the previous comment won't fit in the page because it uses too much notation...

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