Definition (Path connected)

A Topological Space $(X,\mathscr{T})$ is path connected if, for $x_{1},x_{2}\in X$, there exists a continuous map $\gamma:[0,1]\to X$ for which $$\gamma(0)=x_{1}\text{ and }\gamma(1)=x_{2}$$

Definition (Locally path connected)

A Topological Space $(X,\mathscr{T})$ is locally path connected if, for each $x\in X$ and for each Neighbourhood $\mathcal{U}$ of $x$, there exists a neighbourhood $\mathcal{V}$ of $x$ such that if $x_{1},x_{2}\in \mathcal{V}$ then there exists a continuous map $\gamma:[0,1]\to \mathcal{U}$ for which $$\gamma(0)=x_{1}\text{ and }\gamma(1)=x_{2}$$