Locally Path Connected Space

Definition

Locally Path Connected At a Point

Consider a Topological Space $(X,\mathcal{T})$ and a point $p\in X$. The Topological Space is locally path connected at a point $p$ if every Open Set containing $p$ contains a path connected Open Set which contains $p$. $\forall U\in \mathcal{T}\text{s.t.}U\ni p,\exists \text{path connected}C\in \mathcal{T}\text{s.t.}p\in C\subset U$

Or, equivalently, the Topological Space is locally path connected at a point $p$ if $p$ has a Local Basis consisting of path connected open sets.

Locally Path Connected Space

Consider a Topological Space $(X,\mathcal{T})$ The Topological Space is locally path connected If every point in $X$ is locally path connected. $\forall p\in X,\forall U\in \mathcal{T}\text{s.t.}U\ni p,\exists \text{path connected}C\in \mathcal{T}\text{s.t.}p\in C\subset U$

Or, equivalently, the Topological Space is locally path connected if it has a basis consisting of path connected open sets.

Facts

A Topological Space $(X,\mathcal{T})$ is locally connected if and only if for each open subset $U$ of the space $X$, each Path Component $P_U$ of the subset $U$ is an Open Set. $\forall U\in \mathcal{T},P_U\in \mathcal{T}$

If a Topological Space $(X,\mathcal{T})$ is connected and locally path connected, then it is path connected