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 locally connected.

Bibliography

1

S. Willard, General Topology, Addison-Wesley, Publishing Company, 1970.

2

J.G. Hocking, G.S. Young, Topology, Dover Pubs, 1988, republication of 1961 Addison-Wesley edition.