# 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 locally connected.

## References

• 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.
Title connected im kleinen ConnectedImKleinen 2013-03-22 15:59:00 2013-03-22 15:59:00 Mathprof (13753) Mathprof (13753) 8 Mathprof (13753) Definition msc 54D05