separated
Definition Suppose and are subsets of a topological space . Then and are separated provided that
where is the closure operator (http://planetmath.org/Closure) in .
Properties
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1.
If are separated in , and is a homeomorphism, then and are separated in .
Examples
-
1.
On , the intervals and are separated.
-
2.
If , then the open balls and are separated (proof.) (http://planetmath.org/WhenAreBallsSeparated).
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3.
If is a clopen set, then and are separated. This follows since when is a closed set.
Remarks
References
- 1 J.L. Kelley, General Topology, D. van Nostrand Company, Inc., 1955.
- 2 G.J. Jameson, Topology and Normed Spaces, Chapman and Hall, 1974.
Title | separated |
---|---|
Canonical name | Separated |
Date of creation | 2013-03-22 15:16:34 |
Last modified on | 2013-03-22 15:16:34 |
Owner | matte (1858) |
Last modified by | matte (1858) |
Numerical id | 15 |
Author | matte (1858) |
Entry type | Definition |
Classification | msc 54-00 |
Classification | msc 54D05 |