Algebra

Stone–Čech compactification One of the proof of the Stone–Čech compactification is to consider the continuous functions from a given topological space $X$ to $[0, 1]$. Crux 1: constructing $[0, 1]^{C}$ Let $C$ be the space of all continuous functions from $X$ to $[0, 1]$, consider $[0, 1]^{C}$, there is a natural map from $X$ to $[0, 1]^{C}$: for each $x \in X$, define $\phi(x) = f \mapsto f(x)$. With product topology $\phi$ is continuous....

I am reviewing some fundamental algebra, and I just learnt something beautifully suggesting the connection of ring theory and geometry, which made me eager to learn some algebraic geometry (which I was not very interested when I was in undergrad). It is an exercise in Aluffi’s book1. Lets’ jump into it. The problem Let $K$ be a compact topological space, and let $R$ be the ring of continuous real-valued functions on $K$, with addition and multiplication defined pointwise....

Warning You need to know some basic group theory terminology to appreciate(I hope you do) the following content. Burnside’s lemma Here is the statement of the Burnside’s lemma: Let $G$ be a group that actions on a set $X$. Denote $X^{g}$ the set of fixed points of $g$ i.e. $\{x \in X | g \cdot x = x\}$, then the number of orbits of the action is equal to $\dfrac{1}{|G|}\sum\limits_{g \in G} |X^g|$....