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).

References

[a1]	J. van Mill, "Infinite-dimensional topology, prerequisites and introduction" , North-Holland (1988)