Room for Forever Mozart and Balarka Sen

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Balarka Sen

20:48

sup

Forever Mozart

i proved

that if $X$ is connected and every point is a cut point

and $Y$ is a compact space

and $X$ is dense in $Y$

and $Y\setminus X$ is connected

then $Y$ has NO cut points

Balarka Sen

hrm

Forever Mozart

so for example $X=\mathbb R$ and $Y=S^1$

Balarka Sen

cut point is a pt removing which disconnects the space, yeah?

Forever Mozart

yes

Balarka Sen

20:50

k. understand your example

that's really cool

Forever Mozart

yes now we need to think of some related problems

I started this room to collect ideas

Balarka Sen

sounds fun

Forever Mozart

Open Problem 1:

If $X$ is as above, then does $X$ have a compactification w/ no cut pts?

this is difficult

because consider this example

let $S$ be connected and not locally compact

Balarka Sen

yeah for nice spaces it's just one-pt compactification

Forever Mozart

exactly

but take a countable infinite subset of $\beta S \setminus S$

call it $C$.

Balarka Sen

20:55

jeez

Forever Mozart

Now let $X=(\beta S)\setminus C$

Then $X$ is a connected space but there is no $Y$ such that $Y\setminus X$ is connected

because $C$ maps onto every $Y\setminus X$

so $Y\setminus X$ is countable, and therefore not connected

so you may not be able to apply my theorem to X

Balarka Sen

right

Forever Mozart

it seems strange

you could also investigate compactifications of cut(n) spaces

cut(n) space is a space such that every $n$ points disconnects it but no smaller set does

so $S^1$ is a cut(2) space

like, does every cut(1) space have a cut(2) compactification?

and in general cut(n) space has cut(m) compactification what is the relationship btwn n and m

this has not been studied

Balarka Sen

hmm

interesting questions

Forever Mozart

basically I have a 2 page "paper" and need more stuff for it :)

it seems related to graph theory somehow

Balarka Sen

21:04

oh speaking of

do you know an example of a T1 space such that every point disconnects it into 3 pieces?

Forever Mozart

yes, there is even a metric example

Balarka Sen

what's that

Forever Mozart

but somehow there is no separable example

you start with $I=[0,1]$

then to each point you attach another $I$

and you continue $\omega$ many times

so that a point in the space is something with $n$ coordinates

it tells you where to go along the $I$'s

Balarka Sen

Forever Mozart

then the metric is like the taxicab metric

Balarka Sen

21:08

yeah this is something like the example Mike gave

Forever Mozart

remove any point and it should have 3 connected pieces

Balarka Sen

agreed

Forever Mozart

wait

is that right

oh yes its right

Balarka Sen

yep

cool

Forever Mozart

and you can do something similar to get any number of pieces

but this is not separable

Balarka Sen

21:10

what about the rational points in all the intervals

damn uncountable

Forever Mozart

yeah

now I do not know if this space $X$ has a compactification with $Y\setminus X$ connected

Balarka Sen

oh dear dear

Forever Mozart

because it is also not locally compact

so it's possible that this would be a good counterexample

Balarka Sen

yeah. i am just grossed out by the thought of compactifying this

Forever Mozart

yep

well you could simply take $X$ and the set of all infinite sequences

and find a way to topologize that

like the Cantor tree leads to the Cantor set

Balarka Sen

21:14

yeah, just adjoin the end

should be very possible. neat idea

Forever Mozart

if its a counterexample, it would solve a big open problem

see, every separable metric space $X$ has a $Y$ such that $Y\setminus X$ is connected

but for metric spaces in general this is open

Balarka Sen

i see

Forever Mozart

that's a 2008 result

i dont know what other problems to consider

hmm

what if you take that set $X=\beta S\setminus C$

Balarka Sen

i gotta finish this computation i am doing tho

Forever Mozart

and attach an arc to each remaining point in $\beta S\setminus S$

you can make a space with all cut points

and no connected $Y\setminus X$

thats an idea

but you would have to make the arcs shrink to length $0$ everywhere

Balarka Sen

21:24

i dunno much about the stone-cech compactification

Forever Mozart

its big

and not first countable

so its a little difficult to think of attaching things to it

Balarka Sen

fair nuff

Forever Mozart

I'm gonna think about this example for a while

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