connected im kleinen

A topological space $X$ is connected im kleinen at a point $x$ if every open set $U$ containing $x$ contains an open set $V$ containing $x$ such that if $y$ is a point of $V$, then there is a connected subset of $U$ containing $\{x,y\}$.
Another way to say this is that $X$ is connected im kleinen at a point $x$ if $x$ has a neighborhood base of connected sets (not necessarily open).

A locally connected space is connected im kleinen at each point.

A space can be connected im kleinen at a point but not locally connected at the point.

If a topological space is connected im kleinen at each point, then it is locallyconnected.