Apparently the definition of local compactness is not standard. Compactness is obviously a global property of a topological space, so it makes sense to define it locally. Second-countable spaces have countable bases (a global phenomenon), and first-countable spaces are the local version of that, where each point has a countable local base. Being connected is a global property, and being locally connected at a point means every neighborhood is a supserset of some connected set containing the point.
So surely there's no ambiguity as to what a locally compact space should be, right?
I haven't been doing anything that's worth a blog post. I couldn't find a satisfying explanation as to why the coin flip sequence from a previous post corresponds to the Catalan number sequence, but I still plan to make that. After the wave equation, I'd like to make a blog post about how the heat equation is derived.
I've been recently trying out new topics. Steganography looks like a very fun one. I can certainly understand hiding an image within the 4 least-significant bits of each pixel, and I can sort of understand how hiding them in the DCT of a jpeg image works, but anything that's remotely more advanced looks like dark magic.
Similarly, I'd love to learn more about information theory and error correction. I can understand the simple version of parity bits and Reed-Solomon, but anything more advanced like elliptic curves is beyond me. Perhaps I should look for a good book.
I've always known the wave equation since even before taking a proper PDE lecture, from browsing the internet and watching videos, but I never saw how it came to be derived. Like, why does this describe a wave?
$$\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$$
I decided to read Stein and Shakarchi's book on Fourier Analysis (as it is part one of a tetralogy, and I feel like I really need more insight on this topic), and a nice derivation is found right in the beginning.
Alright, time to make a blog post and ignore the massive hiatus between this and the last post. I was watching a streamer making a coin toss bet, where they wanted heads to win. As soon as it was tails, he said "okay, best of 3". It went on heads, but then tails again, to which he continued "okay, best of 5". He then got heads twice, which means heads won the best of 5, and he was content.
My question was: If we kept going, potentially forever, is the chance of heads winning approaching 100%? How fast/slow is it approaching if so? I plan to answer these questions in this post.
There are lots of equivalent statements to the axiom of choice, Zorn's lemma being a famous example. Another interesting one is the fact that every vector space has a basis. We can always find a set of vectors which are linearly independent and generate the entire space.
The proof that every vector space has a basis is simple enough if you already understand Zorn's lemma. You take all the sets which have linearly independent vectors, and the union of any chain of those sets will give a set of linearly independent vectors, so there must exist a maximal linearly independent set, which will be a basis.
Interestingly though, this is an equivalence, we can prove that if every vector space has a basis, then the axiom of choice holds. However, if you try to look for a proof anywhere on the internet, every single possible source will say "This was proven by Andreas Blass and you can see it in this pdf". He did this in 1984 and since then, not a single person bothered to make an alternate proof? Or rewrite it in a more digestible way?
Just learned something cool. If you take any spaces which are connected ($X$ and $Y$), their product is very easily proven to be connected. However, if we take any proper subsets $A \subseteq X$ and $B \subseteq Y$, then
$$(X \times Y) \setminus (A \times B)$$
is also connected. It seemed like a very strong statement at first, but intuitively it makes sense, since this set above is like a "grid", cause if you draw any two proper subsets of $\R$ for example and imagine the set $\R^2 \setminus (A \times B)$, the result will be a grid in some way (unless they're empty). A fun example: $\Q$ and $\R \setminus \Q$ are totally disconnected, yet $\R^2 \setminus \Q^2$ is connected.
I've been casually thinking about that idea I had about making translations for math terms in toki pona from that other post, and I've been taking inspiration from Japanese, since their structure of combining kanji to make new words is basically the challenge I'd have to face.
After looking at some interesting ones (such as "differentiation" being 微分, literally "delicate + part", which is cute), I went on wiktionary to see some of the more mysterious ones, such as "dimension" being 次元 (lit. "next + origin"), and while I was expecting some sort of historical etymology combinations, all it said was "Appeared in (...) “Vocabulary of mathematical terms in English and Japanese” of 1889 as a translation of English dimension.".
To unite topology and group theory, we can define a topological group as a group $(G, \ast)$ with a topology such that $(x, y) \mapsto x \ast y$ and $x \mapsto x\inv$ are continuous functions. Only requiring the first to be continuous is not enough.
Some sources like Munkres additionally requires it to be $T_1$, which seemed arbitrary at first, why not Hausdorff instead? It turns out that (1) being Hausdorff (2) being $T_1$ (3) $\{e\}$ being a closed set; are all equivalent statements for topological groups. So one interesting way to prove some space is Hausdorff is to define a group operation on the space which is continuous and its inverse is continuous too, then prove that $\{e\}$ is closed. This is probably useless.
I just read about quotient spaces, and I'm quite fascinated to have learned that those "images" that I would see online which involve "visualizing" some two-dimensional shape like a torus as a unit square with some funky business going on at the border are actually quite simple to formalize using basic topology and equivalence classes.
The idea is that if we define an equivalence class on a topological space $(X, \tau)$, then there is a unique topology on $X/{\sim}$ such that a subset of equivalence classes is open if and only if its union (which is a subset of $X$) is open in $X$. So when we take a unit square and write "arrows" at the border, what we're really doing is describing visually what the equivalence relation is!
I'm too lazy to put a picture here, I just wanted to share my joy.
Youtube's algorithm recently blessed me with a channel called Zundamon's Theorem. It's a japanese channel about maths where the hosts are actually two characters which are vocal synthesizer characters (basically like Miku, I'm not sure if I can say it's like vocaloid but for speech instead of singing).
It's very cute and the creator actually adds english subs for most videos, which is impressive. Hopefully this will help me learn math terms in japanese such as 定理 (it means theorem and it surprisingly sounds like theorem, probably a coincidence). Hopefully that'll make youtube recommend me more japanese math channels.
