A topological space $(X,\tau)$ is locally connected iff $\forall x\in X$ ,$\exists \{v_{i} : i\in I\}$ a familly of connected neighberhoods of $x$ such that $\forall u$ neighberhood of $x$ $\exists i\in I$ such that $v_{i}\in u$
Show that the n-sphere $\mathbb S^{n}\subseteq \mathbb R^{n+1}$ (Product topology) is locally connected >
One idea is to fixe an element $x=(x_{1},...,x_{n+1})\in \mathbb S^{n}$ and then consider the familly $\mathbb B=\{S^{n}\bigcap \prod_{i=1}^{n+1}]a_{i},b_{i}[\}$ such that $x\in \prod_{i=1}^{n+1}]a_{i},b_{i}[$, we have $\forall B\in \mathbb B$, $B$ is a neighberhood of $x$ in $\mathbb S^{n}$, also $\forall v$ neighberhood of $x$ in $\mathbb S^{n}$, $v$ contains some $B$. Now I need to show that $B$ is connected, and I need some help with that please.