2020-12-26

Misc 0

This post has a lot of interesting information about green threads:

https://graphitemaster.github.io/fibers/

I kind of like the idea of using green threads for Iku, that programming language I've been wanting to work on for a while. The implementation of this seems a bit complicated though.

Topological Groups

If $G$ is a Topological Group, then the connected component $C$ containing the identity $e$ is a normal subgroup of $G$ .

First, note that for any $g \in G$ , $gC$ is also a connected component. If $gx, gy \in gC$ , then $x, y \in C$ . This means they share a connected subspace $A$ . The image $gA$ is also a connected subspace, since the action of $g$ is a continuous function. Evidently, $gA$ contains $gx$ and $gy$ . This means that any two points of $gC$ share a connected subspace.

A similar proof shows that $Cg$ is also a connected component.

It's easy to see that $C$ is a subgroup.

First, $e \in C$ , by assumption.

Second, we need to show that $a, b \in C \implies ab^{-1} \in C$ . Because $e \in C$ , we have $ab^{-1} \in ab^{-1}C$ . Since $a = ab^{-1}b$ , and $b \in C$ , we have $a \in ab^{-1}C$ as well. But since $ab^{-1}C$ is a connected component, i.e. equivalence class, and it contains $a$ , it must be equal to $C$ , since $C$ is an equivalence class containining $a$ . Since $ab^{-1}C = C$ , $ab^{-1}$ must be in $C$ .

To see that $C$ is normal, we show that $gC = Cg$ . For this, note that $gC$ and $Cg$ are both connected components, as we showed earlier. But, since they both contain $g$ , they must be equal.

$\square$

Font Rendering

For some reason I've gotten somewhat curious about font rendering as a potential project. It falls under the "software I use without really understanding, for sure".

Here's some links I got on Discord:

https://gankra.github.io/blah/text-hates-you/ https://crates.io/crates/fontdue