Some traits for S125 Knaster-Kuratowski fan

[prabau]

Provide a justification for some of the traits of S125 that were just referencing the General Reference Chart of S&S.
Remove some redundant traits.

[prabau]

P55, P66: not sure what you mean. The justification is based on corresponding meta-properties. That's what we usually do in these cases. We don't explicitly repeat the metaprops here, they are used implicitly.

[Moniker1998]

@prabau I'm not asking justification of that

[prabau]

@Moniker1998 can you be more specific about what you want to see?

[Moniker1998]

Okay let's take an example.

The closed set $L(c)\cap X\subseteq X$ contains a closed set homeomorphic to {S27}.
And {S27|P55}.

@prabau what's the justification that it contains such closed set

[prabau]

P66: I need to think more about why $[0,1]\setminus\mathbb Q$ is homeomorphic to the irrational numbers.
Must be something simple.

[Moniker1998]

@prabau I mean there's a topological characterization that you can use, but I don't know why you want to prove it

[prabau]

Explaining #1830 (comment) a little more for P55:
the idea is to focus on one of the "rays" L(c) of the fan. Given any non-horizontal line in the plane, projecting it to the y-axis (= taking the second coordinate of evey point) is a homeomorphism $h$. And restricting that homeomorphism to a subset $A$ gives a homeomorphism to $h(A)$.
So $X\cap L(c)$ is homeomorphic to $[0,1/2]\cap\mathbb Q$ (with the subspace topology induced from $\mathbb R$).
All that is pretty basic.

The claim of no isolated point follows from that. And $X\cap L(c)$ is infinite and countable by construction (since $c$ was chosen in $E$).

But all of the above is kind of obvious if one thinks about it geometrically: $X\cap L(c)$ is obtained by taking all the points of the segment $L(c)$ that have rational second coordinate.

What do you think?

Then apply Sierpinski's topological characterization of $\mathbb Q$.
(One could instead use something simpler not involving Sierpinski's result. Will see if I find something already there in mathse. Also need to look for something there for P66.)

[Moniker1998]

@prabau

Explaining #1830 (comment) a little more for P55:
the idea is to focus on one of the "rays" L(c) of the fan. Given any non-horizontal line in the plane, projecting it to the y-axis (= taking the second coordinate of evey point) is a homeomorphism
h. And restricting that homeomorphism to a subset A gives a homeomorphism to h(A).

Yes.

So X∩L(c) is homeomorphic to [0,1/2]∩Q (with the subspace topology induced from R). All that is pretty basic.

This one is a jump in logic. Not basic. The rest is basic.

You should be aware that when you take rays like that, rationals appear and disappear from the line. So it all needs a justification.

[prabau]

There is no jump in logic here. The projection of $L(c)$ onto the second coordinate is a homeomorphism. (Reminder: $L(c)$ is just a slanted ordinary line segment.) I think you have agreed to that, right? So the restriction of that projection to $X\cap L(c)$ is a homeomorphism to its image. And that image is $[0,1/2]\cap\mathbb Q$. So $X\cap L(c)$ is homeomorphic to $[0,1/2]\cap\mathbb Q$. All that is basic.

What is not basic is that $[0,1/2]\cap\mathbb Q$ is homeomorphic to $\mathbb Q$. That requires some reference to something.

Is there anything with the above that you disagree with?

[Moniker1998]

@prabau oh. I think I had some very weird misunderstanding but yeah I see it now.
Yeah I guess that's it. Sorry, I was kind of confused

I've confused it with how the coordinates can be both rational, irrational, only one rational etc. in unpredictable ways

[Moniker1998]

@prabau I think the justification is okay then. But we need to maybe point out how those things are homeomorphic, and give references to characterizations

[Moniker1998]

Then apply Sierpinski's topological characterization of Q. (One could instead use something simpler not involving Sierpinski's result. Will see if I find something already there in mathse. Also need to look for something there for P66.)

I think direct proofs for specific spaces are not that much simpler, better to not bother

[prabau]

I will make some change to clarify something.