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# 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.

Keywords:

connected, locally connected

Type of Math Object:

Definition

Major Section:

Reference

## Mathematics Subject Classification

54D05*no label found*

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