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Definition Suppose and are subsets of a topological space . Then and are separated provided that
where
is the closure operator in .
- If
are separated in , and
is a homeomorphism, then and are separated in .
- On
, the intervals and are separated.
- If
, then the open balls and are separated (proof.).
- If
is a clopen set, then and
are separated. This follows since
when is a closed set.
The above definition follows [1]. In [2], separated sets are called strongly disjoint sets.
- 1
- J.L. Kelley, General Topology, D. van Nostrand Company, Inc., 1955.
- 2
- G.J. Jameson, Topology and Normed Spaces, Chapman and Hall, 1974.
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"separated" is owned by matte. [ full author list (5) ]
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(view preamble)
Cross-references: closed set, clopen set, open balls, intervals, homeomorphism, topological space, subsets
There are 13 references to this entry.
This is version 12 of separated, born on 2005-05-17, modified 2006-05-24.
Object id is 7064, canonical name is Separated.
Accessed 3276 times total.
Classification:
| AMS MSC: | 54-00 (General topology :: General reference works ) | | | 54D05 (General topology :: Fairly general properties :: Connected and locally connected spaces ) |
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Pending Errata and Addenda
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