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Homeseparated

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# separated

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

$\begin{array}[]{ccc}\overline{A}\cap B&=&\emptyset,\\ A\cap\overline{B}&=&\emptyset,\end{array}$ |

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 $d(x,y)\geq r+s$, 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.

# 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.

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## Mathematics Subject Classification

54-00*no label found*54D05

*no label found*

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