# Locally path-connected space

A topological space $X$ in which for any point $x \in X$ and any neighbourhood $O _ {x}$ of it there is a smaller neighbourhood $U _ {x} \subset O _ {x}$ such that for any two points $x _ {0} , x _ {1} \in U _ {x}$ there is a continuous mapping $F : I \rightarrow O _ {x}$ of the unit interval $I = [ 0 , 1 ]$ into $O _ {x}$ with $f ( 0) = x _ {0}$ and $f ( 1) = x _ {1}$. Any locally path-connected space is locally connected. Any open subset of a locally path-connected space is locally path-connected. A connected locally path-connected space is a path-connected space.

Locally path-connected spaces play an important role in the theory of covering spaces. Let $p : ( \widetilde{X} , \widetilde{x} _ {0} ) \rightarrow ( X, x _ {0} )$ be a covering and let $Y$ be a locally path-connected space. Then a necessary and sufficient condition for a mapping $f : ( Y , y _ {0} ) \rightarrow ( X , x _ {0} )$ to admit a lifting, that is, a mapping $g : ( Y , y _ {0} ) \rightarrow ( \widetilde{X} , \widetilde{x} _ {0} )$ such that $f = p \circ g$, is that

$$f _ {\#} ( \pi _ {1} ( Y , y _ {0} ) ) \ \subset p _ {\#} ( \pi _ {1} ( \widetilde{X} , \widetilde{x} _ {0} ) ) ,$$

where $\pi _ {1}$ is the fundamental group. If $X$ is a locally simply-connected (locally $1$- connected, see below) space and $x _ {0} \in X$, then for any subgroup $H$ of $\pi _ {1} ( X , x _ {0} )$ there is a covering $p : ( \widetilde{X} , \widetilde{x} _ {0} ) \rightarrow ( X , x _ {0} )$ for which $p _ {\#} (( \widetilde{X} , \widetilde{x} _ {0} ) ) = H$.

The higher-dimensional generalization of local path-connectedness is local $k$- connectedness (local connectedness in dimension $k$). A space $X$ is said to be locally $k$- connected if for any point $x \in X$ and any neighbourhood $O _ {x}$ of it there is a smaller neighbourhood $U _ {x} \subset O _ {x}$ such that any mapping of an $r$- dimensional sphere $S ^ {r}$ into $U _ {x}$ is homotopic in $O _ {x}$ to a constant mapping. A metric space $X$ is locally $k$- connected if and only if any mapping $f : A \rightarrow X$ from an arbitrary closed subset $A$ in a metric space $Y$ with $\mathop{\rm dim} Y \leq k + 1$ can be extended to a neighbourhood of $A$ in $Y$( the Kuratowski–Dugundji theorem).