CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
A topological space is called locally path-connected if it has a basis of path-connected neighbourhoods. In other words, if for every point $x$ and neighbourhood $V \ni x$, there exists a path-connected neighbourhood $U \subset V$ that contains $x$.
A locally path-connected space is connected if and only if it is path-connected. In any case, the connected components of a locally path-connected space are the same as its path-connected components.
The condition is a necessary assumption in the
in the form of the condition for