A perfect nowhere dense set is a subset of a topological space (usually the real numbers R\mathbb{R}R) that satisfies two seemingly contradictory properties:
Perfect: The set is closed and has no isolated points. Every point in the set is a limit point of other points in the set.
Nowhere dense: The closure of the set has empty interior. In other words, it contains no interval (no nonempty open set).
The most famous example is the Cantor set.
The Cantor Set
Construct it as follows:
Start with the interval [0,1][0,1][0,1].
Remove the open middle third:
(13,23).\left(\frac13,\frac23\right).(31​,32​).
Remove the middle third of each remaining interval:
(19,29),(79,89).\left(\frac19,\frac29\right),\quad \left(\frac79,\frac89\right).(91​,92​),(97​,98​).
Continue forever.
The remaining set is the Cantor set CCC.
Why is it perfect?
It is the intersection of closed sets, so it is closed.
Every point has infinitely many other Cantor points arbitrarily close to it.
There are no isolated points.
Thus, every point is an accumulation point.
Why is it nowhere dense?
Although infinitely many points remain, they are spread so thinly that the set contains no interval whatsoever.
Equivalently,
int⁡(C‾)=int⁡(C)=∅.\operatorname{int}(\overline{C})=\operatorname{int}(C)=\varnothing.int(C)=int(C)=∅. Interesting Properties
The Cantor set is remarkably rich:
✔ Perfect
✔ Nowhere dense
✔ Compact
✔ Totally disconnected
✔ Uncountable
✔ Lebesgue measure 000
These properties make it a cornerstone of real analysis, topology, and fractal geometry.
Why is this surprising?
Intuition often suggests that if a set has no intervals, it should be “small.” But the Cantor set is uncountably infinite—it has the same cardinality as the entire interval [0,1][0,1][0,1]:
∣C∣=∣R∣.|C| = |\mathbb{R}|.∣C∣=∣R∣. So it is “large” in terms of cardinality, yet “small” in terms of measure and topology.
General Definition
In a topological space XXX, a subset AAA is perfect nowhere dense if
A=A‾,A = \overline{A},A=A, A′=A,A’ = A,A′=A, (where A′A’A′ is the set of limit points), and
int⁡(A‾)=∅.\operatorname{int}(\overline{A})=\varnothing.int(A)=∅. Why are perfect nowhere dense sets important?
They appear throughout mathematics:
Baire Category Theorem distinguishes “large” (comeager) sets from nowhere dense ones.
Fractal geometry, where many classic fractals are perfect nowhere dense sets.
Dynamical systems, as invariant Cantor-like attractors.
Functional analysis, where exceptional sets are often constructed as countable unions of nowhere dense sets.
Descriptive set theory, where perfect sets play a central role in classification theorems.
The Cantor set is the canonical example because it demonstrates that a set can be simultaneously:
infinitely detailed,
uncountably large,
have zero length,
contain no intervals,
and yet have every point densely surrounded by other points of the set.ematical set can be closed, uncountably infinite, filled with accumulation points—and still contain no interval.