Continuum hypothesis and fractals

[itssilva]

Wikipedia: "The hypothesis states that the set of real numbers has minimal possible cardinality which is greater than the cardinality of the set of integers" ; i.e., let cardinality of integers = ℵ₀ and cardinality of reals = ℵ_ℝ; then there is no ℵ such that ℵ₀ < ℵ < ℵ_ℝ . But what about fractals? In his book The Fractal Geometry of Nature, Mandelbrot gives a formula for the fractal/Hausdorff dimension of Cantor's set - it's D_F = log 2/log 3 ≈ 0.6309 - as opposed to the topological dimension D_T = 1; clearly, D_F < D_T, and, if you see one of the pictures in the book, it's kinda intuitive that "there's less stuff" in Cantor's set than in, say, the [0; 1] interval - whose cardinality is also ℵ_ℝ -, so whatever D_F is, it seems to be more descriptive of "size" than D_T (actually, Mandelbrot mentions there are technicalities regarding these definitions of dimension, which I'm completely ignorant of). Now, I'll naively assume that the cardinality of ℝ² is (ℵ_ℝ)²; for ℝ², I believe D_F = D_T = 2, but, if I (also naivelly) take the exponent to be (numerically equal to) the fractal rather than the topological dimension, I'd have Cantor's set cardinality ℵ_C = (ℵ_ℝ)^0.6309 < (ℵ_ℝ)¹ . Sooo, if CH is true, what is wrong in my intuition? The technicalities previously mentioned? Or, say, can you prove that there's a bijection between a fractal and ℝ^D_T? (I don't think there's one, self-similarity and "ruggedness" kinda exclude continuity in my POV). (P.S.: I'm prejudiced; regardless of ZFC or any other scheme, I think CH is false, even tho' I ain't no mathematician, and that may be an issue here; just so you know ;P )

 

[jedishrfu]

Can you provide the wikipedia reference you mentioned?

 

[itssilva]

https://en.wikipedia.org/wiki/Continuum_hypothesis

Also, I've quoted Benoit B. Mandelbrot - The Fractal Geometry of Nature, 1st Edition (W. H. Freeman, 1982)

 

[itssilva]

Uh, sorry, I think I glossed over: when I said that for Cantor's dust D_T = 1, I actually meant that that's the initiator of the fractal's - i.e., [0; 1] - D_T; for the dust, D_T = 0, as it is basically a bunch of points. The extrapolation made assumes that, given (D_F)_fractal < (D_T)_initiator , the cardinalities of these two sets oughtta follow the same trend.

 

[rubi]

I don't know about fractals, but you can check that $|C|=|\mathbb R|$ by noting that $|\mathbb R| \leq 2^{\aleph_0}$ (for instance because of the Dedekind cut construction) and since every element of $C$ can be written as $\sum \frac{a_n}{3^n}$ with $a_n\in\{0,2\}$, we have $|C|=2^{\aleph_0}$. Since $C\subseteq\mathbb R$, we have $|C|=|\mathbb R|$.

 

[mathman]

In general dimensionality (as long as it is finite) and cardinality have no relationship.

 

[itssilva]

I don't know about fractals, but you can check that $|C|=|\mathbb R|$ by noting that $|\mathbb R| \leq 2^{\aleph_0}$ (for instance because of the Dedekind cut construction) and since every element of $C$ can be written as $\sum \frac{a_n}{3^n}$ with $a_n\in\{0,2\}$, we have $|C|=2^{\aleph_0}$. Since $C\subseteq\mathbb R$, we have $|C|=|\mathbb R|$.

Thanks for the rep; though I think I get the essential message here, it still feels odd to state that the Cantor set C is same cardinality than ℝ , not by your construction but rather concept: while, in your notation, |[0; 1]| = |ℝ|, if we "subtract" from [0; 1] all reals but, say, those of the form y = 0.xx...(finite # x's)...x000... , we'd have |S = {y}| = |ℤ|; if CH is correct, then there is no "smooth" way to go between these two extremes, which is uncomfortable. You can access any real in [0; 1] by pratically any number of appropriate limiting procedures, but the ones that lead to C are more restricted by the construction of the set itself, so it might be wrong but at least feels sensible enough that |ℤ| = |S| < |C| < |ℝ| by furthering the definition of |set| ; for instance, it sure feels sensible that S ⊆ ℝ and |S| = |ℝ| ; but, is it enough? Does one always imply the other, no matter the structure, topology, continuity (or lack of), etc., of any arbitrary S? Maybe I'm only talkin' BS here, but try addressing this more down-to-earth thing: find a one-to-one mapping between C and ℝ. Maybe it's even trivial, but I can't quite see it.

 

[itssilva]

In general dimensionality (as long as it is finite) and cardinality have no relationship.

Hmm... where could I read about that, is it a general result proven by someone or something? In any case, like I said in the previous rep., I don't see an impossibility in making dimensionality and cardinality have to do with each other - one simply generalizes the definition of the latter, like one generalizes pretty much anything in math according to one's own whims/inclinations, though exactly how, it seems, would be matter for speculation and beyond me.

On a side note: if dimension has nothing to do with cardinality, then |ℝ²| = |ℝ|² is either wrong or accidental; (basic question) how would I figure |ℝ²| without using |ℝ| ? I naively suppose |ℝ^D| = |ℝ|^D , D the topological dimension, is what one invokes, that's why I drew an analogy.

 

[rubi]

Thanks for the rep; though I think I get the essential message here, it still feels odd to state that the Cantor set C is same cardinality than ℝ

Some things in math feel odd, but they are true nevertheless.

Maybe I'm only talkin' BS here, but try adressing this more down-to-earth thing: find a one-to-one mapping between C and ℝ. Maybe it's even trivial, but I can't quite see it.

Well, my proof already establishes that one exists, but here we go: Every real number $x\in(0,1)$ can be written as $x=\sum \frac{a_n}{2^n}$ with $a_n\in\{0,1\}$ (binary expansion). Some numbers have 2 such expansions, for example $(0.100000\ldots)_2 = (0.011111\ldots)_2$. In that case, always choose the non-periodic one. Define $f:(0,1)\rightarrow C$, $f(\sum \frac{a_n}{2^n}) = \sum \frac{2a_n}{3^n}$. This is a bijection between $(0,1)$ and $C$. Then $f(\frac{1}{\pi}\mathrm{arctan}(x)+\frac{1}{2})$ provides a bijection between $\mathbb R$ and $C$.

Edit: Actually this is only an injection $\mathbb R\rightarrow C$, so $C$ might even have more elements than $\mathbb R$. You get a bijection by defining another injection $g:C\rightarrow\mathbb R$, $g(x)=x$ and applying Cantor-Bernstein-Schröder. There is an explicit construction given in the "Book of proof".

 

[FactChecker]

how would I figure |ℝ²| without using |ℝ| ?

The mapping between [0,1] and [0,1]x[0,1] where .d₁d₂d₃d₄d₅d₆... <=> (.d₁d₃d₅d₇d₉d₁₁..., .d₂d₄d₆d₈d₁₀d₁₂...) shows that cardinality does not increase when going from 1 dimension to 2. So |ℝ²| = |ℝ|. And clearly you can do something similar for any finite number of dimensions.

 

[itssilva]

The mapping between [0,1] and [0,1]x[0,1] where .d₁d₂d₃d₄d₅d₆... <=> (.d₁d₃d₅d₇d₉d₁₁..., .d₂d₄d₆d₈d₁₀d₁₂...) shows that cardinality does not increase when going from 1 dimension to 2. So |ℝ²| = |ℝ|. And clearly you can do something similar for any finite number of dimensions.

Fair enough; it also further shows me how uncountable sets are counterintuitive: say you had n evenly spaced integers in the real line, call it V; clearly, |V X V| = |V|² , yet this kind of reasoning fails for |ℝ X ℝ| . I suspect this kinda thing is more important than most physicists make it be: for instance, there are non-perturbative calculations in QCD done over lattices, rather than the whole 4D spacetime. Stuff has been derived in this regime and compared to real life, but I can't but feel uneasy about that because you're throwing away a lot of stuff in the process (maybe one can justify it is irrelevant stuff, but the whole business looks arbitrary). No matter, I digress.

 

[itssilva]

Actually this is only an injection $\mathbb R\rightarrow C$, so $C$ might even have more elements than $\mathbb R$. You get a bijection by defining another injection $g:C\rightarrow\mathbb R$, $g(x)=x$ and applying Cantor-Bernstein-Schröder. There is an explicit construction given in the "Book of proof".

OK, the first sentence is a weird statement (as it sounds eerily Banach-Tarskiesque); I'd expect the possibility of |ℝ| ≥ |C|, but not that |ℝ| ≤ |C| ! Also, your construction in base 2 was a bit strange to me; suppose we expand all reals in I = [0; 1] in, say, decimals - that includes the elements of C , as well - ; since these expansions are unique (right?), a starting point to the mapping I → C would be simply the identity Id: I → I , which is clearly a bijection; but, since we explicitly ditch some numbers (like 0.5) in the construction of C , restricting the image of Id to C would not give a bijection unless you also restrict the domain, which would be saying explicitly that |I| > |C| ; but I don't know, even if you can't reason this way one could possibly cook up a map M that is a bijection - which you did.
Anyway, I'll check out the explicit construction, provided you give me more details of the ref. you cited.

 

[pbuk]

Fair enough; it also further shows me how uncountable sets are counterintuitive: say you had n evenly spaced integers in the real line, call it V; clearly, |V X V| = |V|² , yet this kind of reasoning fails for |ℝ X ℝ| .

It also fails for countably infnite sets - or do you think that there are more rational numbers (pairs of integers) than there are integers?

 

[zinq]

The usual fractal dimension — Hausdorff dimension — of a fractal set in Euclidean space Rⁿ has little to do with its cardinality (which is what the continuum Hypothesis is about). Any subset of Rⁿ with positive Hausdorff dimension must have its cardinality greater than aleph₀. (Note: I've edited this last sentence to correct an earlier mistake.)

One very nice way to think of the Cantor set is that it is topologically equivalent to the countable direct product of a 2-element set with itself:

K ≈ {0, 1}^aleph₀.​

This provides a mapping from the Cantor set onto the real numbers between 0 and 1, by mapping a point (ε₁, ε₂, ε₃, ...) of the Cantor set as above to the binary number having the same sequence of bits. It's easy to show that by ignoring binary expansions ending with all 1's (i.e., of form .?...1111111111...) — which form a merely countable set — this can be made into a bijection. Which shows that

card(K) = 2^aleph₀.​

This can also be seen more directly from the rules about taking powers of cardinalities.

Maybe the easiest way to see this, however, is that K as above is immediately in bijective correspondence with the set of all subsets of the positive integers. (If ε_j = 1, we include j in the subset; otherwise not.) The cardinality of the set of all subsets of any set X (the power set of X) is by definition 2^card(X). And we know from Cantor's theorem that every power set has greater cardinality than the original set, so in particular:

2^aleph₀ > aleph₀.

 

[FactChecker]

Fair enough; it also further shows me how uncountable sets are counterintuitive: say you had n evenly spaced integers in the real line, call it V; clearly, |V X V| = |V|² , yet this kind of reasoning fails for |ℝ X ℝ| . I suspect this kinda thing is more important than most physicists make it be: for instance, there are non-perturbative calculations in QCD done over lattices, rather than the whole 4D spacetime. Stuff has been derived in this regime and compared to real life, but I can't but feel uneasy about that because you're throwing away a lot of stuff in the process (maybe one can justify it is irrelevant stuff, but the whole business looks arbitrary). No matter, I digress.

This may be surprising, but once you realize how it works, it can become fairly intuitive. Even countably infinite sets have the property. If you have a 2-dimensional grid of NxN, you can clearly count them all in order: (0,0), (1,0),(1,1),(0,1), (2,0),(2,1),(2,2),(0,2),(1,2), (3,0),(3,1),(3,2),(3,3),(0,3,),(1,3),(2,3), ...
So |NxN| = |N|. You can do something similar for any finite number of dimensions.

 

[pbuk]

This may be surprising, but once you realize how it works, it can become fairly intuitive. Even countably infinite sets have the property. If you have a 2-dimensional grid of NxN, you can clearly count them all in order: (0,0), (1,0),(1,1),(0,1), (2,0),(2,1),(2,2),(0,2),(1,2), (3,0),(3,1),(3,2),(3,3),(0,3,),(1,3),(2,3), ...
So |NxN| = |N|. You can do something similar for any finite number of dimensions.

I think that's what I said.

 

[Jimster41]

Wikipedia: "The hypothesis states that the set of real numbers has minimal possible cardinality which is greater than the cardinality of the set of integers" ; i.e., let cardinality of integers = ℵ₀ and cardinality of reals = ℵ_ℝ; then there is no ℵ such that ℵ₀ < ℵ < ℵ_ℝ . But what about fractals? In his book The Fractal Geometry of Nature, Mandelbrot gives a formula for the fractal/Hausdorff dimension of Cantor's set - it's D_F = log 2/log 3 ≈ 0.6309 - as opposed to the topological dimension D_T = 1; clearly, D_F < D_T, and, if you see one of the pictures in the book, it's kinda intuitive that "there's less stuff" in Cantor's set than in, say, the [0; 1] interval - whose cardinality is also ℵ_ℝ -, so whatever D_F is, it seems to be more descriptive of "size" than D_T (actually, Mandelbrot mentions there are technicalities regarding these definitions of dimension, which I'm completely ignorant of). Now, I'll naively assume that the cardinality of ℝ² is (ℵ_ℝ)²; for ℝ², I believe D_F = D_T = 2, but, if I (also naivelly) take the exponent to be (numerically equal to) the fractal rather than the topological dimension, I'd have Cantor's set cardinality ℵ_C = (ℵ_ℝ)^0.6309 < (ℵ_ℝ)¹ . Sooo, if CH is true, what is wrong in my intuition? The technicalities previously mentioned? Or, say, can you prove that there's a bijection between a fractal and ℝ^D_T? (I don't think there's one, self-similarity and "ruggedness" kinda exclude continuity in my POV). (P.S.: I'm prejudiced; regardless of ZFC or any other scheme, I think CH is false, even tho' I ain't no mathematician, and that may be an issue here; just so you know ;P )

I'm Reading Manfred Schroeder's "Notes from an Infinite Paradise" (so slowly) and really struggling to get my head around Cantor Dust and continuing fractions - to me there is an intuitive connection between this puzzle and all the when-does-the-wave-not-wave-collapse or not debate as well as the tension between discrete quantum space-time geometry and observer independent continuum.

Not sure if that is sensible or not... which is why I'm struggling. But your question definitely captures my imagination

So the Cantor Space/Set...
Is Totally Disconnected:

• It can only be represented as the union of disjoint sets, a discrete space.

Is Nowhere Dense:

• It's closure has empty interior, all points are boundary points. The reals on the other hand are not nowhere dense so differential calculus is legal.

Is totally Perfect:

• It is a closed set with no isolated points. So how does nowhere dense work if no points are isolated? And if there are no isolated points why isn't differentiation of a Cantor Space legal (or is it)?

Is Compact:

• It Contains all its limit points. How is that possible if it is discrete? Doesn't discrete mean that some point at the shrinking limit of difference between some other two points is - undefined? In other words where are the points at the limit of between the disjoint points that are totally disconnected? They can't be in the set can they?

Is a Metric Space:

• Distances of all its points are defined. This one really breaks my head. How do you tell two self-similar subsets or points in Cantor Space apart? What's the distance between them if they are self-similar? Euclidean distance would have to be zero right? Is the index of iteration somehow included in the notion of measure? Schroeder does the example of shift of decimal representation of ternary numbers to build the "Devil's Staircase". His point I thought is that the representation literally repeats. But If you are counting the number of shifts then it's not really a repetition per se' is it? It's just a similar-looking event n shifts away from the other? Also, what kind of metric space is not differentiable!?

!

Million questions. Sorry.
It is a really bizarre combination of perfect-compact-metric-cardinality and infinitely-recursive-non-linear-diffusion(or evolution) - which just sounds so much like all the Quantum Gravity puzzles.

Also, I blame Lee Smolin, Per Bak, and Steven Strogatz for my combination of confusion and enthusiasm.
http://arxiv.org/pdf/1506.02938v1.pdf See "Real Ensemble Hypothesis" in this one.

Also I took all the definitions and explanations of the properties of Cantor Set/Space from Wiki (after reading Schroeder and others though). That said I very likely botched them in simple ways.

 

[WWGD]

I'm Reading Manfred Schroeder's "Notes from an Infinite Paradise" (so slowly) and really struggling to get my head around Cantor Dust and continuing fractions - to me there is an intuitive connection between this puzzle and all the when-does-the-wave-not-wave-collapse or not debate as well as the tension between discrete quantum space-time geometry and observer independent continuum.

Not sure if that is sensible or not... which is why I'm struggling. But your question definitely captures my imagination

So the Cantor Space/Set...
<Snip>
Is Compact:

• It Contains all its limit points. How is that possible if it is discrete? Doesn't discrete mean that some point at the shrinking limit of difference between some other two points is - undefined? In other words where are the points at the limit of between the disjoint points that are totally disconnected? They can't be in the set can they?

Is a Metric Space:

• Distances of all its points are defined. This one really breaks my head. How do you tell two self-similar subsets or points in Cantor Space apart? What's the distance between them if they are self-similar? Euclidean distance would have to be zero right? Is the index of iteration somehow included in the notion of measure? Schroeder does the example of shift of decimal representation of ternary numbers to build the "Devil's Staircase". His point I thought is that the representation literally repeats. But If you are counting the number of shifts then it's not really a repetition per se' is it? It's just a similar-looking event n shifts away from the other? Also, what kind of metric space is not differentiable!?

!

Million questions. Sorry.
It is a really bizarre combination of perfect-compact-metric-cardinality and infinitely-recursive-non-linear-diffusion(or evolution) - which just sounds so much like all the Quantum Gravity puzzles.

Also, I blame Lee Smolin, Per Bak, and Steven Strogatz for my combination of confusion and enthusiasm.
http://arxiv.org/pdf/1506.02938v1.pdf See "Real Ensemble Hypothesis" in this one.

Also I took all the definitions and explanations of the properties of Cantor Set/Space from Wiki (after reading Schroeder and others though). That said I very likely botched them in simple ways.

Compactness: By Zinq's description as the countable product of the compact set {0,1} with itself (though products of any cardinality will be compact, by Tychonoff's theorem) and

Metrizable: Countable product of metrizable spaces is metrizable, though an uncountable product is not -- 1st countability is lost.

I don't mean to be glib about this; you can see the general proofs and adapt them to the case
of the Cantor set.
Sorry, I don't have time to look at the others now.

 

[Samy_A]

So the Cantor Space/Set...

Is a Metric Space:

• Distances of all its points are defined. This one really breaks my head. How do you tell two self-similar subsets or points in Cantor Space apart? What's the distance between them if they are self-similar? Euclidean distance would have to be zero right? Is the index of iteration somehow included in the notion of measure? Schroeder does the example of shift of decimal representation of ternary numbers to build the "Devil's Staircase". His point I thought is that the representation literally repeats. But If you are counting the number of shifts then it's not really a repetition per se' is it? It's just a similar-looking event n shifts away from the other? Also, what kind of metric space is not differentiable!?

!

I'll address this one.
The Cantor set is a subset of $\mathbb R$. The distance between two points is just the usual distance between two real numbers.

 

[zinq]

"It can only be represented as the union of disjoint sets, a discrete space."

These describe two separate qualities that a space can have. Actually, every nonempty set is the union of disjoint sets.

But the Cantor set is not a "discrete" space, which means that each point has a neighborhood with no other points in it.

Rather, the Cantor set is totally disconnected, meaning that any two points have disjoint neighborhoods. This is true also of the set of rational numbers, but they aren't a discrete set, either.

 

[Samy_A]

Rather, the Cantor set is totally disconnected, meaning that any two points have disjoint neighborhoods. This is true also of the set of rational numbers, but they aren't a discrete set, either.

"Any two points have disjoint neighborhoods" is true for any Hausdorff space.
Probably you meant "Any two points have disjoint neighborhoods that form a partition of the Cantor set".

 

[zinq]

Yes, thank you for catching that mistake, Samy. (I wish I could delete my erroneous post lest it mislead anyone.)

A space X is totally disconnected if its connected components are each a single point. This is exactly the same as saying that every two points x, y of the space lie in two disjoint open sets that partition the entire space:

x ∈ U, y ∈ V, where U and V are disjoint open sets, with U υ V = X.​

Examples of totally disconnected spaces include the Cantor set, the rational numbers, the irrational numbers, and some Julia sets of analytic mappings ℂ → ℂ.

Every discrete set (one whose every point has a neighborhood containing no other point of the set) is totally disconnected, but clearly the converse is not true.

 

[Jimster41]

I'll address this one.
The Cantor set is a subset of $\mathbb R$. The distance between two points is just the usual distance between two real numbers.

Thanks for all the replies,
I sort of realized this yesterday later. I see now where Schroeder says that for every value of x there is a unique value y. So it is only subsets of the Cantor Set (neighborhoods of the Cantor Space?) that are "self-similar" to other subsets and neighborhoods - the distance between those being zero?

As I say it that doesn't seem right... I can see how a geometric representation of a set (or subset) of unique numbers may be identical to another in "appearance" even though the members in each are different. Still the self-similarity, which affects the notion of dimension for the set, must have meaning/effect on the notion of measure and therefor geometry?

"Any two points have disjoint neighborhoods" is true for any Hausdorff space.
Probably you meant "Any two points have disjoint neighborhoods that form a partition of the Cantor set".

any two points have disjoint neighborhoods. I think I can picture that. Do all points have non-disjoint neighborhoods?

Wiki did refer to the Cantor set as "a discrete set" but then wiki does get stuff wrong. It also said "having dimension zero" which I thought was wrong. I thought it had fractal dimension.

 

[Jimster41]

Yes, thank you for catching that mistake, Samy. (I wish I could delete my erroneous post lest it mislead anyone.)

A space X is totally disconnected if its connected components are each a single point. This is exactly the same as saying that every two points x, y of the space lie in two disjoint open sets that partition the entire space:

x ∈ U, y ∈ V, where U and V are disjoint open sets, with U υ V = X.​

Examples of totally disconnected spaces include the Cantor set, the rational numbers, the irrational numbers, and some Julia sets of analytic mappings ℂ → ℂ.

Every discrete set (one whose every point has a neighborhood containing no other point of the set) is totally disconnected, but clearly the converse is not true.

omg... I just got to think about that (bold).
So now I am confused as to how all points in the set are unique? Is that how the self similar effect is described - all self similar points are the same point (kind of makes sense to say it that way) but then I'm confused about how the set can be a mixture of points that are self-similar (repeating) and points that are distinct/unique. Or is it that all points are both? This does seems a bit paradoxical.

Do any points have non-disjoint neighborhoods?
Do all points have non-disjoint neighborhoods?

Definitely fascinating stuff.

 

[Samy_A]

Thanks for all the replies,
I sort of realized this yesterday later. I see now where Schroeder says that for every value of x there is a unique value. So it is only subsets of the Cantor Set (neighborhoods of the Cantor Space?) that are "self-similar" to other subsets and neighborhoods - the distance between those being zero? That doesn't seem right... I can see how a geometric representation of a set (or subset) of unique numbers may be identical to another in "appearance" even though the members in each are different. Still the self-similarity, which affects the notion of dimension for the set, must have meaning/effect on the notion of measure?

The self similarity is not related to the distance between these sets. There is no reason why this distance should be 0.

any two points have disjoint neighborhoods. I think I can picture that. Do all points have non-disjoint neighborhoods?

Yes. Since the whole topological space is by definition open, it is a neighborhood of each element of the topological space.

Wiki did refer to the Cantor set as "a discrete set" but then wiki does get stuff wrong. It also said "having dimension zero" which I thought was wrong. I thought it had fractal dimension.

I don't find these claim on the Wikipedia page for the Cantor set. Are you referring to another Wikipedia page?

EDIT: thanks to @zinq 's post below I now see what you meant. See zinq's post for an answer.

 

[Demystifier]

One should distinguish the measure on the set from the cardinality of the set. Just because one set has a larger measure than the other (e.g. the real interval [0,2] has a larger measure than the real interval [0,1]) does not mean that it has larger cardinality.

 

[zinq]

There are different kinds of dimensions. Hausdorff dimension for a subset of Euclidean space is the commonest kind of dimension used to define a fractional dimension for fractal sets. The Hausdorff dimension d of the standard middle-third Cantor set is easily shown to satisfy

(1/3)^d = 1/2,​

(because 1/2 of it has linear dimension 1/3 of the original) and so

d = log₃(2) = ln(2) / ln(3) ≈ 0.6309...​

This calculation applies to the usual Cantor set, but not necessarily to other Cantor sets that are homeomorphic to this one but not congruent to it.

But there is also a more basic dimension called topological dimension, defined inductively, and which is explained on this page: https://en.wikipedia.org/wiki/Lebesgue_covering_dimension. The topological dimension of the Cantor set (or any set homeomorphic to it) is 0.

(Note: The Wikipedia page on Cantor sets appears to be correct where it refers to a discrete space: It was not referring to the Cantor set, but rather to the countably infinite direct sum of a finite cyclic group with itself.)