MathAtlas

A metric space is a set $X$ together with a function $$ \operatorname{dist}: X \times X \to \R $$ called a metric such that the following laws are satisfied: • (Positivity) $\operatorname{dist}(x,y) \ge…

A homotopy $h : p \simeq q$ between maps $p,q : X \to Y$ is a continuous map \[ h: X \times I \to Y \] such that \[ h(x,0) = p(x), h(x,1) = q(x) \] where $I = [0,1]$ the unit interval.

An equivalence class is a subset of a larger set containing elements that are considered "equivalent" to eachother in the context of a equivalence relation. An equivalence relation is a binary operation denoted $a \s…

Let $$ \gamma_0(t) = (t,0), \quad \gamma_1(t) = (t,t), \quad t \in [0,1]. $$ Both paths go from $(0,0)$ to $(1,1)$. Define a homotopy $H : [0,1] \times [0,1] \to \mathbb{R}^2$ by $$ H(t,s) = (t, s t). $$ Proof th…

A loop is a path \[ f : I \to X \] such that $f(0)=f(1)$

Paths being homotopic defines an equivalence class. Depends on and

Let $f$ be a loop. Let $[f]$ denote the equivalence class under homotopy for $f$. We define $\pi_1(X,x)$ to be the set of equivalence classes of loops that start and end at $x$. We will show that…

The Fundamental Group is indeed a group.

For a path $a : x \to y$, define \[ \gamma[a] : \pi_1(X,x) \to \pi_1(X,y) \] by \[ \gamma[a][f] = [a \cdot f \cdot a^{-1}] \] This is a homomorphism of groups.

Prove that the map $\gamma[a]$ as defined in is indeed a group homomorphism.