Subsets of the Real Line
Subsets of the real number line exhibit a rich variety of behavior. Some sets are small by measure but large by topology. Some are dense but negligible. Some are closed, perfect, and uncountable, yet contain no interval. Some are definable and regular; others require the axiom of choice and are not even Lebesgue measurable.
This post builds a ladder of examples. Each example introduces one new idea: open and closed sets, density, nowhere density, category, \(F_\sigma\) and \(G_\delta\) sets, perfect sets, Cantor-type sets, sets of continuity and differentiability, fractal dimension, and finally non-measurable and non-Borel sets.
The goal is not to list every possible pathology, but to give a working map of the most useful examples.
1. The First Examples: Open, Closed, Neither, Both
A set \(A\subset\mathbb R\) is open if every point of \(A\) has a small open interval around it contained in \(A\). A set is closed if its complement is open, equivalently if it contains all its limit points.
| Set | Type | New Idea |
|---|---|---|
| \(\emptyset\) | open and closed | the trivial clopen set |
| \(\mathbb R\) | open and closed | the other trivial clopen set |
| \(\{x\}\) | closed, not open | singletons are closed |
| finite subset of \(\mathbb R\) | closed, not open | finite unions of closed sets are closed |
| \((0,1)\) | open, not closed | basic open interval |
| \([0,1]\) | closed, not open | compact interval |
| \((0,1]\) | neither open nor closed | half-open behavior |
| \([0,1)\cup(2,3]\) | neither open nor closed | mixed endpoint behavior |
These examples establish the first warning: not open does not mean closed, and not closed does not mean open.
2. Discrete Sets And Limit Points
A point \(x\) is a limit point of \(A\) if every open interval around \(x\) contains a point of \(A\) different from \(x\). A set is discrete if each of its points is isolated from the other points of the set. A point \(x\in A\) is isolated in \(A\) if there is an \(\varepsilon>0\) such that \((x-\varepsilon,x+\varepsilon)\cap A=\{x\}\). Thus \(x\) belongs to the set, but no other points of the set lie sufficiently close to it.
| Set | Type | New Idea |
|---|---|---|
| \(\mathbb Z\) | closed and discrete | infinite closed discrete subset of \(\mathbb R\) |
| \(\{1/n:n\ge 1\}\) | not closed | a set can omit its limit point |
| \(\{0\}\cup\{1/n:n\ge 1\}\) | compact, countable | adding the missing limit point makes the set compact |
| \(\{1/n:n\ge 1\}\cup\{2+1/n:n\ge 1\}\) | compact after adding two limits | multiple accumulation points |
The set \(\{1/n:n\ge 1\}\) is not closed because \(0\) is a limit point not belonging to the set. The set \(\{0\}\cup\{1/n:n\ge 1\}\) is closed and bounded, hence compact by Heine-Borel.
3. Dense Sets, Nowhere Dense Sets, And Boundaries
A set \(A\) is dense in \(\mathbb R\) if every nonempty open interval meets \(A\). The interior of \(A\), denoted \(A^\circ\) or \(\operatorname{int}A\), is the largest open set contained in \(A\). Equivalently, \(x\in A^\circ\) iff there is an \(\varepsilon>0\) such that \((x-\varepsilon,x+\varepsilon)\subset A\). A set is nowhere dense if the closure \(\overline A\) has empty interior. The boundary of \(A\) is \(\partial A=\overline A\setminus A^\circ\). A set \(A\subset\mathbb R\) is hollow if it contains no nonempty open interval.
| Set | Dense? | Nowhere Dense? | Boundary |
|---|---|---|---|
| \(\mathbb Q\) | yes | no | \(\mathbb R\) |
| \(\mathbb R\setminus\mathbb Q\) | yes | no | \(\mathbb R\) |
| \(\mathbb Z\) | no | yes | \(\mathbb Z\) |
| \(\{0\}\) | no | yes | \(\{0\}\) |
| \((0,1)\) | no | no | \(\{0,1\}\) |
| Cantor set \(C\) | no | yes | \(C\) |
The rationals are dense, but they are still very small in other senses: countable, meagre, and measure zero. Density alone is therefore a weak notion of largeness.
A set with dense complement has no interior. A dense set with dense complement has boundary all of \(\mathbb R\). That is why \(\partial\mathbb Q=\mathbb R\).
4. Measure Zero, Full Measure, And Almost Everywhere
A set \(A\subset\mathbb R\) has Lebesgue measure zero if, for every \(\varepsilon>0\), it can be covered by countably many open intervals whose total length is less than \(\varepsilon\). A property holds almost everywhere if it fails only on a set of measure zero.
| Set | Measure | New Idea |
|---|---|---|
| finite set | \(0\) | points have no length |
| countable set | \(0\) | countable unions of null sets are null |
| \(\mathbb Q\) | \(0\) | dense null set |
| \(\mathbb R\setminus\mathbb Q\) | full measure | deleting countably many points changes no length |
| Cantor set | \(0\) | uncountable null set |
| Smith-Volterra-Cantor set | positive | nowhere dense need not mean null |
The Cantor set is the first major surprise: it is uncountable but has measure zero. Cardinality and measure are different notions of size.
5. First Category, Second Category, And Comeagre Sets
A set is meagre, or of first category, if it is a countable union of nowhere dense sets. A set is comeagre in \(\mathbb R\) if its complement is meagre. A set is of second category if it is not meagre.
| Set | Category | New Idea |
|---|---|---|
| \(\{x\}\) | meagre | points are category-small |
| countable set | meagre | countable unions of singletons |
| \(\mathbb Q\) | meagre | dense but category-small |
| \(\mathbb R\setminus\mathbb Q\) | comeagre | topologically large |
| \((0,1)\) | second category | open intervals are not meagre |
| Cantor set | meagre | closed uncountable small set |
The Baire category theorem says that \(\mathbb R\) is not meagre in itself. More generally, no complete metric space is a countable union of nowhere dense sets.
Category and measure are independent languages of largeness. A set can be large by category and small by measure, or small by category and large by measure.
6. \(F_\sigma\) And \(G_\delta\) Sets
An \(F_\sigma\) set is a countable union of closed sets. A \(G_\delta\) set is a countable intersection of open sets.
| Set | Borel Type | Reason |
|---|---|---|
| countable set | \(F_\sigma\) | union of singleton closed sets |
| \(\mathbb Q\) | \(F_\sigma\) | countable union of singletons |
| \(\mathbb R\setminus\mathbb Q\) | \(G_\delta\) | complement of an \(F_\sigma\) set |
| open set in \(\mathbb R\) | \(F_\sigma\) | approximate from inside by closed sets |
| closed set in \(\mathbb R\) | \(G_\delta\) | approximate from outside by open sets |
| Cantor set | closed and \(G_\delta\) | every closed subset of \(\mathbb R\) is \(G_\delta\) |
For example,
\[ (0,1)=\bigcup_{n\ge 2}[1/n,1-1/n], \]
so \((0,1)\) is \(F_\sigma\). Also,
\[ [0,1]=\bigcap_{n\ge 1}(-1/n,1+1/n), \]
so \([0,1]\) is \(G_\delta\).
The rationals are \(F_\sigma\) but not \(G_\delta\). If \(\mathbb Q\) were \(G_\delta\), then \(\mathbb Q\) and \(\mathbb R\setminus\mathbb Q\) would be disjoint dense \(G_\delta\) sets, contradicting the Baire category theorem.
7. Perfect Sets And Cantor Sets
A set is perfect if it is closed and has no isolated points. Every nonempty perfect subset of \(\mathbb R\) is uncountable. In fact, every nonempty perfect subset of \(\mathbb R\) has cardinality continuum.
| Set | Perfect? | Interior | Measure |
|---|---|---|---|
| \([0,1]\) | yes | nonempty | \(1\) |
| Cantor set \(C\) | yes | empty | \(0\) |
| Smith-Volterra-Cantor set | yes | empty | positive |
| \(\{0\}\cup\{1/n:n\ge 1\}\) | no | empty | \(0\) |
The standard middle-thirds Cantor set is obtained from \([0,1]\) by repeatedly removing open middle thirds. It is closed, compact, perfect, nowhere dense, uncountable, and null.
The Cantor set introduces the difference between having many points and having length. It has as many points as the whole real line, but total length zero.
8. Fat Cantor Sets And The Smith-Volterra-Cantor Set
A fat Cantor set is a Cantor-like closed nowhere dense set with positive Lebesgue measure. The standard Smith-Volterra-Cantor set is built by removing open intervals from \([0,1]\) with total length \(1/2\). The remaining set is closed, perfect, nowhere dense, and has measure \(1/2\).
This example shows that nowhere dense is a topological notion, not a measure-theoretic one. A set can have empty interior and still have positive measure.
More generally, for any \(a\in[0,1)\) we can construct a closed nowhere dense perfect subset of \([0,1]\) with measure \(a\).
9. Dense Open Sets Of Small Measure
Enumerate the rationals as \(q_1,q_2,\dots\). Given \(\varepsilon>0\), define
\[ G=\bigcup_{n\ge 1}\left(q_n-\varepsilon 2^{-n-2},q_n+\varepsilon 2^{-n-2}\right). \]
The set \(G\) is open because it is a union of open intervals. It is dense because it contains every rational. But its measure is less than \(\varepsilon\).
This construction shows that open and dense does not imply large measure. Topological largeness and measure-theoretic largeness are genuinely different.
10. Dense \(G_\delta\) Null Sets
A dense \(G_\delta\) null set is a countable intersection of dense open sets, but has Lebesgue measure zero. Such sets are topologically large and measure-theoretically small.
We can construct one by choosing open dense sets \(G_N\) with \(m(G_N)<2^{-N}\) and setting
\[ G=\bigcap_{N\ge 1}G_N. \]
Then \(G\) is dense \(G_\delta\) by the Baire category theorem, but \(m(G)=0\) because \(G\subset G_N\) for every \(N\).
This is the basic Oxtoby phenomenon: a residual set can be null.
11. Liouville Numbers
A real number \(x\) is a Liouville number if, for every \(N\ge 1\), there are integers \(p,q\) with \(q\ge 2\) such that
\[ 0<\left|x-\frac pq\right|<\frac{1}{q^N}. \]
Liouville numbers are real numbers that can be approximated by rationals extraordinarily well. Every Liouville number is transcendental, by Liouville’s theorem on Diophantine approximation.
The set of Liouville numbers is dense \(G_\delta\) and has Lebesgue measure zero. Hence it is large by category and small by measure. Its complement has full measure but is meagre.
| Set | Category | Measure |
|---|---|---|
| Liouville numbers | comeagre | zero |
| non-Liouville numbers | meagre | full |
This is one of the cleanest standard examples showing that category-typical and measure-typical are not the same.
12. Normal Numbers
A real number is normal in base \(b\) if, in its base-\(b\) expansion, every block of \(k\) digits occurs with limiting frequency \(b^{-k}\). For example, in a normal decimal number each digit occurs with frequency \(1/10\), each pair of digits with frequency \(1/100\), and so forth.
The set of numbers normal in base \(b\) has full Lebesgue measure. In fact, almost every real number is normal in every integer base \(b\ge 2\).
Normality is a measure-theoretic typicality result. Explicit examples exist, such as Champernowne’s number in base \(10\),
\[ 0.123456789101112131415\cdots, \]
but most familiar constants, such as \(\pi\) and \(e\), are not known to be normal in any base.
13. Badly Approximable Numbers
A real number \(x\) is badly approximable if there is a constant \(c>0\) such that
\[ \left|x-\frac pq\right|>\frac{c}{q^2} \]
for all integers \(p\) and \(q\ge 1\).
Equivalently, \(x\) has bounded partial quotients in its continued fraction expansion. The golden ratio is the archetypal badly approximable number.
The set of badly approximable numbers has Lebesgue measure zero, is meagre, but has Hausdorff dimension \(1\). This example introduces a third notion of size: fractal dimension. A set can have measure zero but still be dimensionally large.
14. Hausdorff Dimension Examples
Hausdorff dimension refines the crude distinction between countable, null, positive measure, and full measure. It measures how efficiently a set can be covered by small intervals.
| Set | Lebesgue Measure | Hausdorff Dimension |
|---|---|---|
| countable set | \(0\) | \(0\) |
| Cantor set | \(0\) | \(\log 2/\log 3\) |
| badly approximable numbers | \(0\) | \(1\) |
| interval \([0,1]\) | positive | \(1\) |
| fat Cantor set | positive | \(1\) |
The Cantor set has dimension \(\log 2/\log 3\), reflecting the construction: at each step, two copies survive, each scaled by a factor \(1/3\).
Badly approximable numbers show that dimension \(1\) does not imply positive measure.
15. Sets Of Continuity Points
For a function \(f:\mathbb R\to\mathbb R\), the set of points where \(f\) is continuous is always a \(G_\delta\) set. Therefore, the set of discontinuities is always an \(F_\sigma\) set.
Examples:
| Function | Continuity Set | Discontinuity Set |
|---|---|---|
| constant function | \(\mathbb R\) | \(\emptyset\) |
| indicator \(1_{\mathbb Q}\) | \(\emptyset\) | \(\mathbb R\) |
| indicator \(1_C\) of Cantor set | \(\mathbb R\setminus C\) | \(C\) |
| Thomae function | \(\mathbb R\setminus\mathbb Q\) | \(\mathbb Q\) |
Thomae’s function is defined by
\[ f(x)= \begin{cases} 1/q, & x=p/q\text{ in lowest terms},\\ 0, & x\notin\mathbb Q. \end{cases} \]
It is continuous exactly at the irrational points and discontinuous exactly at the rational points. Thus a countable dense set can occur as the discontinuity set of a real function.
16. Sets Of Discontinuities
A classical theorem says that a set \(D\subset\mathbb R\) is the set of discontinuities of some real-valued function if and only if \(D\) is an \(F_\sigma\) set.
This gives a functional interpretation of the Borel class \(F_\sigma\).
| Candidate Set \(D\) | Can Be A Discontinuity Set? | Reason |
|---|---|---|
| \(\emptyset\) | yes | continuous functions |
| finite set | yes | step functions |
| \(\mathbb Q\) | yes | Thomae’s function |
| Cantor set | yes | indicator of Cantor set |
| \(\mathbb R\) | yes | Dirichlet function \(1_{\mathbb Q}\) |
| \(\mathbb R\setminus\mathbb Q\) | no | not \(F_\sigma\) |
The irrational numbers cannot be exactly the set of discontinuities of a real-valued function, because they are not \(F_\sigma\).
17. Monotone Functions And Countable Discontinuities
If \(f:\mathbb R\to\mathbb R\) is monotone, then its set of discontinuities is at most countable. Each discontinuity is a jump discontinuity.
This provides a strong contrast with arbitrary functions. Arbitrary functions can be discontinuous on any \(F_\sigma\) set. Monotone functions are much more rigid.
Examples:
| Function | Discontinuities |
|---|---|
| \(f(x)=x\) | none |
| \(f(x)=1_{\{x>0\}}\) | \(\{0\}\) |
| distribution function of a discrete random variable | countable jumps |
| Cantor function | none |
The Cantor function is continuous, nondecreasing, constant on the removed middle-third intervals, and increases from \(0\) to \(1\) entirely on the Cantor set. It is a useful bridge between topology, measure, and probability.
18. Differentiability Sets
For arbitrary functions, differentiability is less clean than continuity. For monotone functions, however, Lebesgue’s differentiation theorem gives a powerful result: every monotone function is differentiable almost everywhere.
Examples:
| Function Type | Differentiability Behavior |
|---|---|
| \(C^1\) function | differentiable everywhere, derivative continuous |
| Lipschitz function | differentiable almost everywhere |
| absolutely continuous function | differentiable almost everywhere and recovered by integrating its derivative |
| monotone function | differentiable almost everywhere |
| Cantor function | derivative \(0\) almost everywhere, but total increase \(1\) |
| Weierstrass-type function | continuous everywhere, differentiable nowhere |
The Cantor function is especially important. It is continuous and monotone, hence differentiable almost everywhere, and its derivative is \(0\) almost everywhere. Nevertheless, the function rises from \(0\) to \(1\). The increase is carried by the Cantor set, which has measure zero.
19. Nowhere Differentiable Continuous Functions
A continuous function can fail to be differentiable at every point. The classical Weierstrass function gives such an example. It has the form
\[ f(x)=\sum_{n=0}^{\infty}a^n\cos(b^n\pi x) \]
for suitable constants \(0<a<1\) and odd integer \(b\) with \(ab\) large enough.
The existence of nowhere differentiable continuous functions was historically surprising. In the space \(C[0,1]\) with the uniform norm, nowhere differentiable functions are not exceptional by category: they form a comeagre set. Thus a “typical” continuous function is nowhere differentiable in the Baire category sense.
20. Sets With The Baire Property
A set \(A\subset\mathbb R\) has the Baire property if there is an open set \(G\) such that the symmetric difference \(A\triangle G\) is meagre. In words, \(A\) is open up to a meagre error.
All Borel sets have the Baire property. All analytic sets also have the Baire property. But some sets constructed using the axiom of choice do not have the Baire property.
The Baire property is the category analogue of Lebesgue measurability. It says that a set is regular when viewed through the lens of category.
21. Borel Sets And The Borel Hierarchy
The Borel sets are obtained from open intervals by repeatedly taking countable unions, countable intersections, and complements. They form the smallest \(\sigma\)-algebra containing the open sets.
The first levels are:
| Class | Description | Example |
|---|---|---|
| open | union of open intervals | \((0,1)\) |
| closed | complement of open | \([0,1]\) |
| \(F_\sigma\) | countable union of closed sets | \(\mathbb Q\) |
| \(G_\delta\) | countable intersection of open sets | \(\mathbb R\setminus\mathbb Q\) |
| \(F_{\sigma\delta}\) | countable intersection of \(F_\sigma\) sets | many natural limit-condition sets |
| \(G_{\delta\sigma}\) | countable union of \(G_\delta\) sets | many natural oscillation sets |
Borel sets are highly regular: every Borel subset of \(\mathbb R\) is Lebesgue measurable and has the Baire property.
22. Analytic Non-Borel Sets
An analytic set is a continuous image of a Borel subset of a Polish space. Equivalently, a subset of \(\mathbb R\) is analytic if it is the projection of a Borel subset of \(\mathbb R^2\).
Every Borel set is analytic, but not every analytic set is Borel. Analytic non-Borel sets are important because they lie just beyond the Borel hierarchy while retaining strong regularity properties.
Every analytic subset of \(\mathbb R\) is Lebesgue measurable and has the Baire property. Thus “not Borel” does not automatically mean pathological in the measure-theoretic or category-theoretic sense.
23. Coanalytic Sets And The Projective Boundary
A set is coanalytic if its complement is analytic. A set that is both analytic and coanalytic is Borel, by Souslin’s theorem.
Beyond analytic and coanalytic sets lies the projective hierarchy, formed by repeatedly taking projections and complements. Regularity questions become more delicate there. Under strong set-theoretic axioms, many projective sets are measurable and have the Baire property. Under the axiom of choice alone, pathological sets exist.
This marks the transition from classical real analysis to descriptive set theory.
24. Vitali Sets
Define an equivalence relation on \([0,1]\) by
\[ x\sim y \quad\text{if and only if}\quad x-y\in\mathbb Q. \]
A Vitali set chooses exactly one representative from each equivalence class. The axiom of choice guarantees the existence of such a set.
A Vitali set is not Lebesgue measurable. The reason is that rational translates of the Vitali set are disjoint, but countably many such translates cover \([0,1]\) up to a bounded enlargement. If the Vitali set had measure zero, the union of its rational translates would have measure zero. If it had positive measure, countably many disjoint translates would force infinite measure inside a bounded interval. Both alternatives are impossible.
Vitali sets introduce non-measurability through translation invariance and countable additivity.
25. Bernstein Sets
A Bernstein set \(B\subset\mathbb R\) is a set such that both \(B\) and \(\mathbb R\setminus B\) meet every nonempty perfect subset of \(\mathbb R\).
Equivalently, neither \(B\) nor its complement contains a nonempty perfect set, but neither can avoid one.
Consequences:
| Property | Bernstein Set |
|---|---|
| Borel | no |
| Lebesgue measurable | no, for the standard construction |
| Baire property | no |
| contains a perfect subset | no |
| complement contains a perfect subset | no |
| cardinality | continuum |
Bernstein sets are designed to defeat the perfect-set structure of the real line. They are more topologically pathological than Vitali sets, which are designed around rational translation classes.
26. Hamel Bases
A Hamel basis is a basis of \(\mathbb R\) as a vector space over \(\mathbb Q\). That is, every real number is expressible uniquely as a finite rational linear combination of basis elements.
The existence of a Hamel basis uses the axiom of choice. Any Hamel basis is highly nonconstructive. It is not Borel, not Lebesgue measurable, and does not have the Baire property.
Hamel bases show that an algebraically natural object can be measure-theoretically and topologically wild.
27. Non-Measurable Subgroups
There are subgroups of \(\mathbb R\) under addition that are not Lebesgue measurable. For example, a proper vector subspace of \(\mathbb R\) over \(\mathbb Q\) obtained from a Hamel basis gives such examples.
The key principle is that a measurable subgroup of \(\mathbb R\) is heavily constrained. If it has positive measure, then it must contain an interval around \(0\), and hence must be all of \(\mathbb R\). Therefore a proper measurable subgroup must have measure zero. Many algebraically large proper subgroups cannot have measure zero, so they are non-measurable.
This example connects algebraic structure with measure-theoretic rigidity.
28. The Main Contrasts
The most important lesson is that different notions of size disagree.
| Example | Cardinality | Topology | Category | Measure |
|---|---|---|---|---|
| \(\mathbb Q\) | countable | dense | meagre | \(0\) |
| \(\mathbb R\setminus\mathbb Q\) | continuum | dense \(G_\delta\) | comeagre | full |
| Cantor set | continuum | perfect nowhere dense | meagre | \(0\) |
| Smith-Volterra-Cantor set | continuum | perfect nowhere dense | meagre | positive |
| dense open set of tiny measure | continuum | open dense | second category | tiny |
| Liouville numbers | continuum | dense \(G_\delta\) | comeagre | \(0\) |
| non-Liouville numbers | continuum | dense \(F_\sigma\)-type | meagre | full |
| badly approximable numbers | continuum | thin but rich | meagre | \(0\) |
| Vitali set | continuum | non-regular | no Baire property | non-measurable |
| Bernstein set | continuum | meets every perfect set | no Baire property | non-measurable |
29. A Suggested Teaching Sequence
A nice sequence of examples is:
- \(\emptyset\) and \(\mathbb R\): clopen sets.
- \(\{x\}\): closed point.
- \((0,1)\), \([0,1]\), and \((0,1]\): open, closed, and neither.
- \(\mathbb Z\): infinite closed discrete set.
- \(\{1/n:n\ge 1\}\): missing limit point.
- \(\{0\}\cup\{1/n:n\ge 1\}\): compact countable set.
- \(\mathbb Q\): dense, countable, meagre, null.
- \(\mathbb R\setminus\mathbb Q\): dense \(G_\delta\), comeagre, full measure.
- Cantor set: perfect, nowhere dense, uncountable, null.
- Smith-Volterra-Cantor set: nowhere dense but positive measure.
- Dense open set of tiny measure: topology large, measure small.
- Dense \(G_\delta\) null set: residual but null.
- Liouville numbers: canonical comeagre null set.
- Non-Liouville numbers: full measure but meagre.
- Normal numbers: measure-typical digit behavior.
- Badly approximable numbers: measure zero but full Hausdorff dimension.
- Continuity sets of functions: always \(G_\delta\).
- Discontinuity sets of functions: exactly \(F_\sigma\).
- Monotone functions: only countably many discontinuities.
- Cantor function: continuous singular increase on a null set.
- Weierstrass function: continuous nowhere differentiable behavior.
- Analytic non-Borel sets: beyond Borel but still regular.
- Vitali sets: non-measurability from rational translations.
- Bernstein sets: pathological interaction with perfect sets.
- Hamel bases: algebraic structure with analytic pathology.
30. Closing Summary
The real line carries several different notions of size and regularity: cardinality, density, interior, closure, category, Lebesgue measure, Hausdorff dimension, Borel complexity, definability, and algebraic structure. The interesting subsets of \(\mathbb R\) are interesting because these notions disagree.
The rationals are dense but countable. The Cantor set is uncountable but null. The Smith-Volterra-Cantor set is nowhere dense but has positive measure. The Liouville numbers are comeagre but null. The non-Liouville numbers have full measure but are meagre. The badly approximable numbers have measure zero but full Hausdorff dimension. Analytic non-Borel sets are not Borel but remain measurable. Vitali and Bernstein sets finally leave the regular world altogether.
A good mental model is that no single word – dense, uncountable, measurable, residual, null, perfect, Borel – captures “large” or “small.” Each belongs to a different geometry of the real line.
Glossary
- Interior
- For \(A\subset\mathbb R\), the interior of \(A\), denoted \(A^\circ\) or \(\operatorname{int}A\), is the largest open set contained in \(A\). Equivalently, \(x\in A^\circ\) iff there is an \(\varepsilon>0\) such that \((x-\varepsilon,x+\varepsilon)\subset A\).
- Boundary
- For \(A\subset\mathbb R\), the boundary of \(A\), denoted \(\partial A\), is the set of points where every open interval meets both \(A\) and \(\mathbb R\setminus A\). Equivalently, \(\partial A=\overline A\setminus A^\circ\).
- Isolated Point
- A point \(x\in A\) is isolated in \(A\) if there is an \(\varepsilon>0\) such that \((x-\varepsilon,x+\varepsilon)\cap A=\{x\}\). Thus \(x\) belongs to the set, but no other points of the set lie sufficiently close to it.
- Limit Point
- A point \(x\in\mathbb R\) is a limit point of \(A\) if every open interval around \(x\) contains a point of \(A\) different from \(x\). The point \(x\) need not itself belong to \(A\).
- Hollow Set
- A set \(A\subset\mathbb R\) is hollow if it contains no nonempty open interval. Equivalently, \(A^\circ=\emptyset\). Thus hollow means “empty interior.” The rationals \(\mathbb Q\), the irrationals \(\mathbb R\setminus\mathbb Q\), the Cantor set, and every singleton are hollow subsets of \(\mathbb R\).
- Nowhere Dense
- A set \(A\subset\mathbb R\) is nowhere dense if its closure has empty interior, i.e. \((\overline A)^\circ=\emptyset\). Every nowhere dense set is hollow, but not every hollow set is nowhere dense. For example, \(\mathbb Q\) is hollow, but not nowhere dense, because \(\overline{\mathbb Q}=\mathbb R\).
- Meagre Or First Category
- A set \(A\subset\mathbb R\) is meagre, or of first category, if it is a countable union of nowhere dense sets. Countable sets, the rationals, and the Cantor set are meagre.
- Second Category
- A set \(A\subset\mathbb R\) is of second category if it is not meagre. The real line \(\mathbb R\), every nonempty open interval, and every dense \(G_\delta\) subset of \(\mathbb R\) are second category.
- Comeagre
- A set \(A\subset\mathbb R\) is comeagre if its complement is meagre. Equivalently, \(A\) contains the intersection of countably many dense open sets. The irrational numbers are comeagre, because their complement \(\mathbb Q\) is meagre.
- Comeagre Versus First And Second Category
- In \(\mathbb R\), every comeagre set is second category. Indeed, if both \(A\) and \(\mathbb R\setminus A\) were meagre, then \(\mathbb R\) would be meagre, contradicting the Baire category theorem. However, in a meagre ambient space, a set can be comeagre and meagre relative to that space. For example, in the subspace \(\mathbb Q\), the set \(\mathbb Q\) is comeagre in \(\mathbb Q\) and also meagre as a space.
- Relative Comeagre
- A set \(A\subset X\) is comeagre in a topological space \(X\) if \(X\setminus A\) is meagre in \(X\). The phrase “comeagre” is therefore always relative to an ambient space, unless the ambient space is clear.
- Example: Comeagre And Second Category In \(\mathbb R\)
- The irrational numbers \(\mathbb R\setminus\mathbb Q\) are comeagre in \(\mathbb R\), because \(\mathbb Q\) is meagre. They are therefore second category in \(\mathbb R\).
- Example: Comeagre And First Category In A Subspace
- In the subspace \(X=\mathbb Q\), the whole space \(X\) is comeagre in itself, because \(X\setminus X=\emptyset\) is meagre in \(X\). But \(X=\mathbb Q\) is meagre in itself, because \(\mathbb Q\) is countable and each singleton is nowhere dense in \(\mathbb Q\). Thus “comeagre” does not by itself imply “second category” unless the ambient space is a Baire space such as \(\mathbb R\).
- Smith-Volterra-Cantor Set
- The Smith-Volterra-Cantor set is a closed, perfect, nowhere dense subset of \([0,1]\) with positive Lebesgue measure, usually \(1/2\). It is also called a fat Cantor set. It shows that closed nowhere dense sets need not have measure zero.