I Think There Are Uncountably Many Homeomorphism Types Of Countable Punctured Planes.

I think there are uncountably many homeomorphism types of countable punctured planes.

Curves in the plane with dense punctures (for example, what’s left of the unit circle after you puncture the rational points on the unit circle out of existence) are nowhere path connected in a way that non-convergent Cauchy sequences can pick up, so homeomorphisms ought to send densely punctured curves to other densely punctured curves. Then number and nesting patterns of densely punctured curves would give a nice invariant.

But in particular, after cutting out rational points from circles of integer radius about the origin, we create a bunch of un-punctured rings, only one of which has as its interior an open disk (that’s the middle one).

then simply by puncturing the rational points from a_1 disjoint loops in the first circle, a_2 from the second, a_3 from the third and so on, we can inject the integer sequence (a_n) into the set of homeomorphism classes of punctured planes. (Since countable unions of countable sets of punctures are countable, the resulting construction actually lives in the set of countably punctured planes).

That jeans post has got me thinking, is there a nice way of distinguishing between planes with infinite numbers of holes in them. More formally, can we classify up to homeomorphism infinitely punctured planes easily?

I have a feeling cardinality matters. I'd conjecture that ℝ²\(ℚ×{0}) is not homeomorphic to ℝ²\(C×{0}), where C is the middle third Cantor set. But how could we prove that?

Also I have a feel cardinality isn't the only thing that matters. I reckon density might matter too. For example, is ℝ²\(ℚ×{0}) homeomorphic to ℝ²\(ℕ×{0}). I guess the difference there is that each hole in the second space is isolated whereas those in the first aren't.

It'd be interesting to hear what other people think :))

I’d like to share a couple of highlights of my discord messages today.

7:42 pm

AAAAAAA I HATE INDUCTION

The book my team is using as a reference suppressed a really specific detail of a proof by hiding it as an exercise

And it’s like 8 pages of induction

11:40 pm

Frothing at the mouth ok there are a couple of problems with my induction and the actual proof is like three lines of triangle inequality.

anyway, the moral of the story is that in a δ-hyperbolic geodesic space (one where any point on a side of a triangle is within δ of the other two sides), any 8δ-local geodesic (a path that is preserves distances between any two points within 8δ of each other in the domain) stays uniformly within 2δ of the geodesic connecting its endpoints.

Some days, my problems can be resolved by a simple :%s/<.\{-}>//g

(that’s the vim command that removes html and leaves behind the unformatted text).

Also, piratesexmachine420, you have an excellent pfp. Triangles are good.

This meeting could've been an elaborate and transfixingly beautiful flag semaphore transmission.

Weierstrauss function

my friend sent this gif in the group chat and its just SO FUNNY TO ME for some reason

Hi, I'm

Math enjoyers do not DNI. If you like math you are required to interact with me.

at some point you have to realize that you actually have to read to understand the nuance of anything. we as a society are obsessed with summarization, likely as a result of the speed demanded by capital. from headlines to social media (twitter being especially egregious with the character limit), people take in fragments of knowledge and run with them, twisting their meaning into a kaleidoscope that dilutes the message into nothing. yes, brevity is good, but sometimes the message, even when communicated with utmost brevity, requires a 300 page book. sorry.