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Adherent, Accumulation & Isolated

Click any point on the canvas. The visualization instantly classifies it and shows why — an animated shrinking ball reveals which set points are nearby. Adjust ε to explore manually.

Set
Click to probe · Scroll to adjust ε · The true classification uses the smallest possible ε
50

How to read this

The colored ball around your probe point shows what's inside B(x, ε). Circled dots are set points caught by the ball.

  • Orange ring: set points captured (proves adherence)
  • Teal ring: other set points captured (proves accumulation)
  • Try shrinking ε — if points always remain captured no matter how small, the classification holds for all ε.

Classification

Click a point on the canvas to classify it.
Set point
Adherent
Accumulation
Isolated
None

Adherent

x is adherent to S if every ball around it touches S. Equivalently, there is no gap separating x from S.

x ∈ S̄ ⟺ ∀ε>0 : B(x,ε) ∩ S ≠ ∅

Accumulation

x is an accumulation (limit) point if every ball catches a set point other than x. Points of S crowd around x.

x ∈ S′ ⟺ ∀ε>0 : B(x,ε) ∩ (S ∖ {x}) ≠ ∅

Isolated

x ∈ S is isolated if some ball around it catches no other set points. x is in S but stands alone.

x isolated ⟺ x ∈ S ∧ ∃ε>0 : B(x,ε) ∩ S = {x}