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 |
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Canonical name | ConnectedImKleinen |
Date of creation | 2013-03-22 15:59:00 |
Last modified on | 2013-03-22 15:59:00 |
Owner | Mathprof (13753) |
Last modified by | Mathprof (13753) |
Numerical id | 8 |
Author | Mathprof (13753) |
Entry type | Definition |
Classification | msc 54D05 |