# Misc 0

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: