$\pi_0$ Notation

Question

Let $X$ be a topological space. Then by definition $\pi_0(X)$ = {Path-connected Components of $X$}.

My professor introduced this notation and I'm wondering if it's standard notation or if there's a more standard notation for finding this partition of $X$.

Answer 1

$π_0$, is the set of path-components of $X$, and you can give it the quotient topology. For a very large class of spaces, path components are open and so the quotient topology $π_0$ will be discrete.

For $X$ with base point $b$, we define $π_n(X)$, the $n$-th homotopy group (http://ncatlab.org/nlab/show/homotopy+group), to be the set of homotopy classes of maps $f : S^n → X$ that map the base point $a$ to the base point $b$. For $n\geq1$, the homotopy classes form a group. For $n\geq2$, $\pi_n$ is abelian (see the http://en.wikipedia.org/wiki/Eckmann%E2%80%93Hilton_argument.) We call $π_1$ as the fundamental group.

There is a long exact sequence of homotopy groups $$... → π_n(A) → π_n(C) → π_n(B) → π_{n−1}(A) →... → π_0(C) → 0.$$ Here, $ p: C → B$ is a basepoint-preserving Serre fibration with fiber $A$.

Answer 2

It's pretty standard, yes. You'll also learn about $\pi_1(X)$ and later $\pi_n(X)$, and it fits the pattern. $\pi_0(X)$ is a set, but $\pi_1(X)$ is a group, and $\pi_n(X)$ ends up being abelian groups for $n>1$. For $n\geq 1$, the groups $\pi_n(X)$ are called "homotopy groups." Not sure what they call $\pi_0(X)$, when using names.

Technically, the homotopy groups (and $\pi_0$) are actually defined for a pointed topology - $\pi_n(X,x_0)$ for some $x_0\in X$. For $n=0$, the choice of $x_0$ is basically irrelevant. For $n>0$, it is more complicated, but at the very least depends on the path-connected component of $x_0$.