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?
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.
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.
