The difference between the small inductive dimension, the large inductive dimension and the lebesgue covering dimension

Question

I am studying fractals autodidactically (I have never had any topology discipline) and according to Mandelbrot: a fractal has a Hausdorff dimension that exceeds the topological dimension. I understood what Hausdorff's dimension is: it is the value of $s$ (unique) for which a jump between infinity and zero occurs:
$$ \dim_H(X):=\sup\{s:\mathcal{H}^s(F)=\infty\}:=\inf\{s:\mathcal{H}^s(F)=0\} $$
However, I have been trying to understand what the topological dimension is for several days and three concepts always appear to me: the small inductive dimension, the large inductive dimension and the lebesgue covering dimension. I do not understand the definition and the difference between the three and to understand the concept of fratal I just need to understand the concept of topological dimension. Could you explain the concept to me intuitively and with examples and then the mathematical formalism, please?

Answer 1

First off, the term "fractal" does not have a universally agreed upon meaning. I have written about this before on Math SE, so I won't go into that again. I will point out that Mandelbrot walked back from the definition you provide in the second edition of The Fractal Geometry of Nature, so even he was not entirely wed to that notion.

Inductive Dimensions

Regarding the meat of your question, you cite three different topological notions of dimension: the small inductive dimension, the large inductive dimension, and the Lebesgue covering dimension. For the sake of completeness, let us recall the definitions (note: this presentation assumes familiarity with some of the basics of point-set topology, i.e. the definitions of open and closed sets, limit points, boundaries, etc—these definitions are cribbed from James Robinson's Dimensions, Embeddings, and Attractors):

Definition: The small inductive dimension, denoted $\DeclareMathOperator{ind}{ind}\ind(\cdot)$, is defined as follows:

1. $\ind(\varnothing) = -1$,
2. $\ind(X) \le n$ if for every point $p \in X$, $p$ has "arbitrarily small" neighborhoods $U$ with $\ind(\partial U) \le n-1$, where $\partial U$ denotes the boundary of $U$, and
3. $\ind(X) = n$ if $\ind(X) \le n$ but it is not true that $\ind(X) \le n-1$.

The large inductive dimension, denoted $\DeclareMathOperator{Ind}{Ind}\Ind(\cdot)$ is defined similarly, with (2) replaced by

2. $\Ind(X) \le n$ if for every closed set $A \subseteq X$ and each open set $V \subseteq X$ which contains $A$ there exists an open set $U \subseteq X$ such that $$ A\subseteq \overline{U} \subseteq V \qquad\text{and}\qquad \Ind(\partial U) \le n-1. $$

Both of these definitions are pretty gnarly, but they are trying to get at the same essential idea: we can understand the dimension of a set by looking at the boundaries of subsets; the dimension of a space should be greater than the dimension of the boundary of an open subset of the space.

Thinking in a very Euclidean way, an open set in $\mathbb{R}^2$ is (more or less) a disk. The boundary of a disk is a circle, so $\mathbb{R}^2$ should be one dimension greater than a circle. An open set in the circle is (roughly) an interval, and the boundary of an interval consists of two points. Thus the circle should have dimension one greater than a set containing two points. The boundary of a discrete set is empty, and so a collection of discrete points should have dimension one greater than the emptyset. Thus

\begin{align} \DeclareMathOperator{Dim}{dim} \Dim(\mathbb{R}^2) &= 1 + \Dim(\text{a circle}) \\ &= 1 + (1 + \Dim(\text{two points})) \\ &= 1 + (1 + (1 + \Dim(\varnothing))) \\ &= 1 + (1 + (1 + -1))) \\ &= 2, \end{align} where $\Dim$ denotes some appropriate notion of dimension.

The precise definitions of the small and large inductive dimensions try to capture this idea in a more general way, with slight variations in the way in which the open sets of concern are chosen. As is often the case, the jump from intuitive to precise requires a great deal of work.

Covering Dimension

The Lebesgue covering dimension approaches things slightly differently:

Definition: The order of an open covering of a topological space is the largest $n$ such that there exist $n+1$ sets in the covering with non-empty intersection. An open covering $\mathscr{V}$ is a refinement of an open covering $\mathscr{U}$ if every member of $\mathscr{V}$ is contained in some element of $\mathscr{U}$.

A set $A$ has Lebesgue covering dimension less than or equal to $n$, denoted $\Dim_{L}(A) \le n$, if every covering has a refinement of order less than or equal to $n$. $\dim_{L}(A) = n$ if $\dim_{L}(A) \le n$, but not $\Dim_{L}(A) \le n-1$.

Again, there is a pretty intuitive observation about Euclidean space running around behind the scenes. Imagine trying to cover $\mathbb{R}$ by open intervals, and then throwing away as many sets as you can. What you will likely end up with is a collection of intervals which overlap "just a little" at their ends, see the figure.

No matter how gnarly the initial cover is, I can always find a way to "throw away" enough sets so that the remaining sets don't overlap too much. Specifically, no point on the real line will be contained in more than two of the covering intervals.

By contrast, if you attempt to cover the plane $\mathbb{R}^2$ by disks such that no point is contained in more than two disks, you will rapidly find that this is impossible—there must be some points contained in three disks.

Further Reading

Robinson, James C., Dimensions, embeddings, and attractors, Cambridge Tracts in Mathematics 186. Cambridge: Cambridge University Press (ISBN 978-0-521-89805-8/hbk). xii, 205 p. (2011). ZBL1222.37004.

Engelking, Ryszard, Dimension theory. A revised and enlarged translation of ”Teoria wymiaru”, Warszawa 1977, by the author, North-Holland Mathematical Library. Vol. 19. Amsterdam, Oxford, New York: North-Holland Publishing Company. Warszawa: PWN - Polish Scientific Publishers. X, 314p. $ 44.50; Dfl. 100.00 (1978). ZBL0401.54029.

Hurewicz, W.; Wallman, H., Dimension theory., 165 p. Princeton University Press (1941). ZBL67.1092.03.