separated
Definition Suppose A and B are subsets of a topological space X. Then A and B are separated provided that
ˉA∩B=∅,A∩ˉB=∅, |
where ˉA is the closure operator (http://planetmath.org/Closure) in X.
Properties
-
1.
If A,B are separated in X, and f:X→Y is a homeomorphism, then f(A) and f(B) are separated in Y.
Examples
-
1.
On ℝ, the intervals (0,1) and (1,2) are separated.
-
2.
If d(x,y)≥r+s, then the open balls Br(x) and Bs(y) are separated (proof.) (http://planetmath.org/WhenAreBallsSeparated).
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3.
If A is a clopen set, then A and A∁ are separated. This follows since ˉS=S when S 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 |