Forever Mozart's Room

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Forever Mozart

04:51

@BrianM.Scott There was a problem with my construction. Now I'd like to answer this simpler question: Is there a $G_\delta$ in $[0,1]^2$ that is dense in every horizontal line but contains no vertical line?

Intuitively do you think yes or no?

Martin Sleziak

05:34

@ForeverMozart Maybe I am missing something, but can't you get such set simply by omitting the diagonal from the square? I.e., $[0,1]^2\setminus\Delta$.

I have omitted exactly one point from each horizontal line, so the intersection with the horizontal line is dense in the horizonal section.

I have omitted one point from each vertical line, therefore no vertical line is contained in the set.

Forever Mozart

Ah, yes :)

now let me make it more difficult

suppose we want to contain no vertical interval?

Martin Sleziak

ok, this will eliminate the simple solution I suggested above

Forever Mozart

so that the space is totally disconnected

hmm

what if you remove a diagonal $\omega$-many times?

would the horizontal sections be dense?

so I mean to remove shited (by rationals) copies of the diagonal

Martin Sleziak

You mean removing set like $\Delta_d=\{(x,y)\in I^2; x-y=d\}$?

Right.

Forever Mozart

yeah that should work

Martin Sleziak

05:40

So you are removing $D=\bigcup\limits_{d\in\mathbb Q} \Delta_d$.

Forever Mozart

yes

Martin Sleziak

Intersection of $D$ with any vertical/horizontal line is just "shifted copy" of the set $\mathbb Q$.

Forever Mozart

yeah so that works right?

Martin Sleziak

Intersection of the complement with any line is just "shifted copy" of $\mathbb R\setminus\mathbb Q$.

Forever Mozart

yes

Martin Sleziak

05:41

Which means that both $D$ and the complement are dense in each v/h line. (But do not contain any interval.)

Unless I missed something.

And $D$ is clearly a countable union of closed sets. So it is $F_\sigma$ and the complement is $G_\delta$.

Forever Mozart

yeah so it works

Martin Sleziak

To me it seems like this set works.

Forever Mozart

thanks for this example

Martin Sleziak

Well, you came up with the example after I suggested $I\setminus\Delta$ for the easier problem.

Forever Mozart

so do you see how this relates to my original problem?

Martin Sleziak

05:45

Sorry, I did not have the time to have a closer look at your paper (in progress) on Lelek's fan.

That is what you meant by your original problem right?

Forever Mozart

Ok, well what I want is a dense connected $G_\delta$ in the Cantor fan, such that the vertex point is a dispersion point

Such a set would have to hit every cross section densely, and contain no interval along the lines of the fan

Martin Sleziak

Cantor fan is just rays from one point to Cantor set?

Forever Mozart

of course connectedness makes things difficult

Yes

Martin Sleziak

Is it called Cantor teepee by some people? Or is that a different space?

Forever Mozart

I think Cantor teepee is another name for the KK fan

where you take lines of rationals/irrationals

whereas Cantor fan has the entire lines

Martin Sleziak

05:48

Forever Mozart

see, the KK fan doesn't work

cause it has closed copies of rationals

it cant be $G_\delta$

Martin Sleziak

Ok, I think that I see that problems of finding such subset in $I\times I$ and $C\times I$ and Cantor fan are somehow similar.

I am sorry, but I will have to leave now. (Time to go to work.)

Forever Mozart

Ok, well thanks for your input, see you later

Martin Sleziak

Have a nice day!

Martin Sleziak

06:39

@ForeverMozart Did you manage to get electronic version of the paper you asked about here: Help finding paper from the 1920's?

Forever Mozart

Yes, the person who responded scanned and sent me a copy

emailed it to me

Martin Sleziak

Forever Mozart

unfortunately this example cannot help much

because $G_\delta$ graphs are nowhere dense

but still it is good to know about the history of strange connected spaces

Martin Sleziak

I just found out that our university has access to that paper from Springer website. So I wanted to offer you the same thing (e-mailing it to you).

But since you already have it, it's moot point now.

Forever Mozart

well thanks anyway. I have access to lots of sites through my university library, but not Springer.

Martin Sleziak

06:44

We have quite good access to Springer and Elsevier journals.

It's not that good with some other publishers. (For example, de Gruyter. Or the access to papers on jstor we have is rather limited.)

But overall I'd guess the situation here is not that bad.

Ok, I just dropped in to ask about this. I should get to marking some tests and homework.

See you later!

Forever Mozart

I can find 90% of papers just by scrounging around online

but occasionally theres a rare one

ok bye

Martin Sleziak

I think in some cases Sci-hub might help. But it seems to be down at the moment.

It is a site similar to Library Genesis but for journals.

Bye!

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