A point of a set that has a neighborhood containing no other points of the set; equivalently a point in the set that is not an accumulation point.

Principle

Isolation means local discreteness: the set does not cluster at the point because one can find an open region around the point free of other members of the set.

Demonstration

In the set {0} ∪ {1/n : n ∈ N} on the real line, each 1/n is isolated by a small enough interval that excludes other 1/m, whereas 0 is not isolated because points 1/n accumulate at 0.

Misapplication

Assuming isolated points are negligible in topological arguments can be wrong: they can change compactness or connectedness claims (finite isolated points can break connectedness or affect closure calculations).

Consequence

Identifying isolated points separates discrete components from cluster structures, simplifies local analyses (e.g., defining functions by values at isolated points), and clarifies sequence convergence properties.

Reversal

The reverse concept is accumulation: points where every neighborhood contains other set points. Where isolation holds, accumulation fails locally.

Boundary

Applies in any topology; excludes notions of isolation measured by distance magnitude unless the topology is metric. An isolated point may still be a boundary point of the set if the neighborhood intersects the complement as well.

Semantic Tension

Tension arises with 'discrete point' and 'atom' in measure theory: an isolated point is a topological notion and does not automatically imply positive measure or algebraic atomicity.

Synthesis

An isolated point is a member of a set that stands alone in some neighborhood: a locally discrete element that marks the absence of clustering at that location.