A topological space is **locally connected** if every point has a neighborhood basis of connected open subsets.

A property of topological spaces is termed **hereditary**, **subspace-hereditary**, if every subset of a topological space having the property also has the property (when the subset is endowed with the subspace topology).

The (Alexandroff) topological spaces in which any point has the least open neighborhood are hereditarily locally connected.

Another example of hereditarily locally connected topological space is $L=\mathbb N\cup\{\infty\}$ with opens $U\subseteq L$ upward closed subsets such that $U\neq\{\infty\}$. Note that this space is not an Alexandroff space, since $\infty$ does not have the least open neighborhood.

I wonder what classes of topological spaces have the hereditarily locally connectedness property. Is there any study available regarding hereditarily locally connectedness?