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Note: We look at a few historical notes about the beginning of dimension theory in topology. We can find this way some arguments which led mathematicians to introduce new types of dimensions by means of functions.

Then we take a look at fractal geometry and provide some information about desirable properties of dimensions.

The following is a verbatim excerpt from Theory of Dimensions - Finite and Infinite by Ryszard Engelking. Please note that many sections in the book contain detailed and much more indepth historical notes whereas the few snippets I present here hardly touch the surface of the information which can be found in the book.

From Section 1.1 Dimension theory of separable metric spaces The dimension of simple geometric objects is one of the most intuitive mathematical notions. There is no doubt that a segment, a square and a cube have dimension $1$, $2$ and $3$ respectively. The necessity of a precise definition of dimension became obvious only when it was established that a segment has exactly as many points as a square (G. Cantor 1878), and that a square has a continuous parametric representation on a segment, i.e., that there exist continuous functions $x(t)$ and $y(t)$ such that the points of the form $(x(t),y(t))$ fill out a square when $t$ runs through a segment (G. Peano 1890).

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First and foremost, the question arose ... whether the $n$-cube $I^n$ and the $m$-cube $I^m$ are homeomorphic if $n

e m$; clearly a negative answer was expected. The theorem that $I^n$ and $I^m$ are not homeomorphic if $n

e m$ was proved by Brouwer in [1911]. It suggests itself to try to prove this theorem by definining a function $d$ which assigns to every space a natural number, the dimension of the space, so that homeomorphic spaces have equal dimensions and $d(I^n)=n$. It was none too easy, however, to discover such functions; the search for them gave rise to dimension theory. ... A decisive step towards the definition of dimension was made by Poincarè in [1903], where he observed that dimension is related to the notion of separation and could be defined inductively. Poincarè called attention to the simple fact that solid bodies can be separated by surfaces, surfaces by lines, and lines by points. In [1912] he defined $n$-dimensional continuum as one which can be divided into several parts by means of cuts along $(n-1)$-dimensional continua. ...

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The first definition of a dimension function was given by Brouwer in 1913, where he defined a topological invariant of compact metric spaces, called Dimensionsgrad and proved that Dimensionsgrad of the $n$-cube $I^n$ is equal to $n$. Following Poincarè's suggestion, the definition is inductive and refers to the notion of a cut: ...

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... Referring to the second part of Lebesgue's paper [1911], Mazurkievicz showed in [1915] that for every continuous parametric representation $f$ of the square $I^2$ on the interval $I$, some fibres of $f$ have cardinality at least $3$, and proved that every continuum $C\subset R^2$ whose interior in $R^2$ is empty can be represented as a continuous image of the the Cantor set under a mapping with fibres of cardinality at most $2$. These results led him to define the dimension of a compact metric space $X$ as the smallest integer $n$ with the property that the space $X$ can be represented as the continuous image of a closed subspace of the Cantor set under a mapping $f$ such that $|f^{-1}|\leq n+1$ for every $x\in X$. It was proved later that this definition is equivalent to the definition of the small inductive dimension, but Mazurkievicz's paper had no influence on the development of dimension theory. The definition of the small inductive dimension $ind$ was formulated by Urysohn in [1922] and Menger in [1923]; ...

Here is an example which shows that sometimes decades can pass before a dimension to specific spaces could be associated.

From Section 1.4 Dimension theory of separable metric spaces Totally disconnected spaces were introduced by Sierpiński in [1921], hereditarily disconnected spaces - by Hausdorff in [1914], and punctiform spaces - by Janiszewski in [1912]. ... The first example of a totally disconnected space which is not zero-dimensional was given by Sierpiński in [1921]; Sierpiński's space is a completely metrizable subspace of the plane. The first example of a hereditarily disconnected space which is not totally disconnected was also given by Sierpiński in [1921]; this space is also a completely metrizable subspace of the plane. The first example of a punctiform space which is not hereditarily disconnected was described by Sierpiński in [1920]; this space is a connected subspace of the plane. An example of a complete metrizable punctiform and connected subspace of the plane was given by Mazurkiewicz in [1920]. ... A dimension function associated with the class of totally disconnected spaces was defined by O'Connor and Rogers in [1992].

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From Section 1.6 Definitions of the large inductive dimension and the covering dimension. Metric dimension The first separation theorem, ... , suggests a modification in the definition of the small inductive dimension - replacing the point $x$ by a closed set $A$. In this way we are led to the notion of the large inductive dimension $Ind$, defined for all normal spaces. Both dimensions coincide in the realm of separable metric spaces. They diverge, however, in the class of metric spaces and also in the class of compact spaces, ...

Add-on [2016-05-25]:

Now we take at look at the world of fractals and the way how dimensions can be specified. We will also see, that there are sometimes many different technical terms for one and the same kind of dimension. The following is from Fractal Geometry by Kenneth Falconer.

From chapter 3 Alternative definitions of dimensions Hausdorff dimension, ..., is the principal definition of dimension that we shall work with. However, other definitions are in widespread use, and it is appropriate to examine some of these and their inter-relationship. ...

Fundamental for a definition of dimension

Fundamental to most definitions is the idea of measurement at scale $\delta$. For each $\delta$, we measure a set in a way that ignores irregularities of size less than $\delta$, and we see how these measurements behave as $\delta \rightarrow 0$. For example, if $F$ is a plane curve, then our measurement, $M_\delta(F)$, might be the number of steps required by a pair of dividers set at length $\delta$ to traverse $F$. A dimension of $F$ is then determined by the power law (if any) obeyed by $M_\delta(F)$ as $\delta \rightarrow 0$. If \begin{align*} M_\delta(F)\sim c\delta^{-s}\tag{1} \end{align*} for constants $c$ and $s$, we might say that $F$ has dimension $s$, with $c$ regarded as the $s$-dimensional length of $F$. Taking logarithms \begin{align*} \log M_\delta(F)\simeq \log c - s\log \delta \end{align*} in the sense that the difference of the two sides tends to $0$ with $\delta$, and \begin{align*} s=\lim_{\delta \rightarrow 0}\frac{\log M_\delta(F)}{-\log \delta}\tag{2} \end{align*} ... There may be no exact power law for $M_\delta(F)$, and the closest we can get to (2) are the lower and upper limits. For the value of $s$ given by (1) to behave like a dimension, the method of measurements needs to scale with the set, so that doubling the size of $F$ and at the same time doubling the scale at which measurement takes place does not affect the answer, ...

Properties a dimension should have

There are no hard and fast rules for deciding whether a quantity may reasonably be regarded as dimension. There are many definitions that do not fit exactly into the above, rather simplified szenario. The factors that determine the acceptability of a definition of a dimension are recognized largely by experience and intuition. In general one looks for some sort of scaling behaviour, a naturalness of the definition in the particular context and properties typical of dimensions such as those discussed below. A word of warning: as we shall see, apparently similar definitions of dimension can have widely differing properties. It should not be assumed that different definitions give the same value of dimension, even for nice sets. Such assumptions have led to major misconceptions and confusion in the past. It is necessary to derive the properties of any dimension from its definition. ... What are the desirable properties of a dimension? Those derived in the last chapter for Hausdorff dimension are fairly typical.

Monotonicity. If $E\subset F$ then $\dim_H E\leq \dim_H F$.

Stability. $\dim_H(E \cup F)=\max(\dim_H E,\dim_H F)$.

Countable stability. $\dim_H\left(\bigcup_{i=1}^{\infty} F_i\right)=\sup_{1\leq i<\infty}\dim_H F_i$.

Geometric invariance. $\dim_H f(F)=\dim_H F$ if $f$ is a transformation of $\mathbb{R}^n$ such as a translation, rotation, similarity or affinity.

Lipschitz invariance. $\dim_H f(F)=\dim_H F$ if $f$ is a bi-Lipschitz transformation.

Countable sets. $\dim_H F=0$ if $F$ is finite or countable.

Open sets. If $F$ is an open subset of $\mathbb{R}^n$ then $\dim_H F=n$.

Smooth manifolds. $\dim_H F =m$ if $F$ is a smooth $m$-dimensional manifold.

All definitions of dimension are monotonic, most are stable, but as we shall see, some common definitions fail to exhibit countable stability and may have countable sets of positive dimension. All the usual dimensions are Lipschitz invariant, and, therefore, geometrically invariant. The open sets and smooth manifolds properties ensure that the dimension is an extension of the classical definition. Note that different definitions of dimension can provide different information about which sets are Lipschitz equivalent.

Box-counting dimensions