Reference :Algebraics Topology by Allen Hatcher
Lemma $1.15$.: If a space $X$ is the union of a collection of path-connected open sets $A_{\alpha}$ each containing the base point $x_0\in X$ and if each intersection $A_{\alpha}\cap A_{\beta}$ is path-connected, then every loop in $X$ at $x_0$ is homotopic to a product of loops each of which is contained in a single $A_{\alpha}$.
Diagram 
From the diagram we can conclude that every loop in $X$ at $x_0$ is homotopic to a product of loops each of which is contained in a single $A_{\alpha}$ and $A_{\beta}$
My confusion: In the last sentence, Why Hachter didn't mention about $A_{\beta}$ ?
My thinking : All loops are contained in both $A_{\alpha}$ and $A_{\beta}$
Best Answer
Here's a somewhat more formal statement.
Suppose we have some set $I$ and a function $A : I \to P(X)$ such that $A(\alpha)$ is a path-connected open neighbourhood of $x_0$ for all $\alpha \in I$. Suppose further that for all $\alpha, \beta \in I$, $A(\alpha) \cap A(\beta)$ is path-connected. Then every loop with base point $x_0$ is homotopic to the product of loops $\ell_1, ..., \ell_n$, such that for each $i$ s.t. $1 \leq i \leq n$, there is some $\alpha_i \in I$ such that the range of $\ell_i$ lies entirely within $A(\alpha_i)$.