Isolated point

In topology , an element of a set is an isolated point if there is a neighborhood of in which (apart from ) no further elements of lie. A point is isolated if and only if there is no accumulation point of . ${\ displaystyle a}$ ${\ displaystyle X}$ ${\ displaystyle a}$ ${\ displaystyle a}$ ${\ displaystyle X}$ ${\ displaystyle a \ in X}$ ${\ displaystyle a}$ ${\ displaystyle X}$

If every point of a topological space is isolated, the space is called discrete .

Examples

The following examples use subsets of real numbers with the usual topology.

There is an isolated point in the crowd .

{\ displaystyle \ {0 \} \ cup [1,2]}

{\ displaystyle 0}

In the set , each of the elements is an isolated point, but is not an isolated point.

{\ displaystyle \ {0 \} \ cup \ {1, {\ tfrac {1} {2}}, {\ tfrac {1} {3}}, \ dots \}}

{\ displaystyle {\ tfrac {1} {n}}}

{\ displaystyle 0}

In the set of natural numbers , all elements are isolated points. So it's a discrete room . ${\ displaystyle \ mathbb {N} = \ {0,1,2, \ dots \}}$

Individual evidence

^ Johann Cigler , Hans-Christian Reichel : Topology. A basic lecture (= BI university paperbacks. 121). Bibliographisches Institut, Mannheim et al. 1978, ISBN 3-411-00121-6 , § 2.3 definition.
↑ Oliver Deiser: Real Numbers. The classical continuum and the natural consequences. 2nd, corrected and enlarged edition. Springer, Berlin et al. 2008, ISBN 978-3-540-79375-5 , chap. 2.1, definition on page 299.

[1] Johann Cigler , Hans-Christian Reichel : Topology. A basic lecture (= BI university paperbacks. 121). Bibliographisches Institut, Mannheim et al. 1978, ISBN 3-411-00121-6 , § 2.3 definition.

[2] Oliver Deiser: Real Numbers. The classical continuum and the natural consequences. 2nd, corrected and enlarged edition. Springer, Berlin et al. 2008, ISBN 978-3-540-79375-5 , chap. 2.1, definition on page 299.