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separated (Definition)

Definition Suppose $ A$ and $ B$ are subsets of a topological space $ X$. Then $ A$ and $ B$ are separated provided that

\begin{displaymath} \begin{array}{ccc} \overline{A}\cap B &=& \emptyset, \ A\cap \overline{B} &=& \emptyset, \end{array}\end{displaymath}
where $ \overline{A}$ is the closure operator in $ X$.

Properties

  1. If $ A,B$ are separated in $ X$, and $ f\colon X\to Y$ is a homeomorphism, then $ f(A)$ and $ f(B)$ are separated in $ Y$.

Examples

  1. On $ \mathbbmss{R}$, the intervals $ (0,1)$ and $ (1,2)$ are separated.
  2. If $ d(x,y)\ge r+s$, then the open balls $ B_r(x)$ and $ B_s(y)$ are separated (proof.).
  3. If $ A$ is a clopen set, then $ A$ and $ A^\complement$ are separated. This follows since $ \overline{S}=S$ when $ S$ is a closed set.

Remarks

The above definition follows [1]. In [2], separated sets are called strongly disjoint sets.

Bibliography

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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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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