We already said that Cantor’s definition had one fault - it was not at all suitable for curves in space. But then what is a surface in space? No one knew. This problem - to determine what curves and surfaces in space are - was put in the summer of 1921 to his twentythree year old student Pavel Samuelovich Urysohn by the venerable Professor Dimitri Fedorovich Yegorov of Moscow University (it is evident that he thought a lot about the mathematical significance of the problem or, as is sometimes said today, of the “dissertability” of the subject - this problem was one of the hardest!)
Urysohn quickly comprehended that Yegorov’s problem was only a special case of a much more general problem: what is the dimension of a geometric figure, i.e., what are the characteristics of the figure which cause us to say that a segment or circumference has dimension 1, a square has dimension 2, and a cube or sphere has dimension 3? Here is what is remembered about this period in the life of P. S. Urysohn by his closest friend, a young doctoral candidate in those days and now an academician, the honorary president of the Moscow Mathematical Society, Pavel Sergeevich Aleksandrov: “… the whole summer of 1921 was spent in trying to find an ‘up-to-date’ definition (of dimension); P. S. shifted his interest from one variant to another, constantly setting up examples showing why this or that variant had to be eliminated. He spent two months totally absorbed in his meditations. At last, one morning near the end of August, P. S. awoke with his now well-known inductive definition of dimension in its final form… That very morning, while we were bathing in the Klyaz’ma, P. S. Urysohn told me about his definition of dimension and there, during the conversation that extended over several hours, outlined a plan for a complete theory of dimension composed of a series of theorems, which were then hypotheses that he did not yet know how to go about proving and which were later proved one after another in the months that followed. I never again either participated in or witnessed a mathematical conversation composed of such a dense flow of new ideas as the conversation of that August morning.
(…)
Let us now discuss more precisely how Urysohn defined the dimension of a geometric figure. A typical zero-dimensional set would be a set consisting of a single point or, in the worst case, of a finite number of points. But in such a set each point has a relative neighborhood with empty boundary - the point itself. This was the property that Urysohn took for his definition of a set of dimension zero
A set F has dimension zero, if each of its points has an arbitrarily small relative neighborhood with empty boundary.
In most cases it is possible to establish that a set has dimension zero by selecting for each point an arbitrarily small ordinary neighborhood whose boundary contains no point of the set F (then the boundary of the relative neighborhood is sure to be empty). But there are zero-dimensional sets situated in three-dimensional space for whose points such ordinary neighborhoods are not available.
The words “arbitrarily small” are inserted in the definition for the following reason. If these were not there, then we could, for instance, find a circle big enough to hold an entire square within it and so that no point of the square would be on the boundary of the circle. So if these words were not in the definition, we would find that the dimension of a square is zero, not two as it really is.
In addition to finite sets, many infinite sets have dimension zero, For example, take the set of points of the \(x\) axis with coordinates \(0, 1, \frac{1}{2}, \frac{1}{3}, …, \frac{1}{n}, …\). It is clear that any point of this set has an arbitrarily small neighborhood that does not contain any points of this set. Only the case of the point \(0\) might cause some doubts. But if we take a neighborhood of radius \(\alpha\), where \(\alpha\) is an irrational number, then no point of the set will occur on the boundary of this neighborhood.
A set F has dimension one, if it is not zero-dimensional and each of its points has an arbitrarily small neighborhood whose boundary intersects the set F in a zero-dimensional set.
It turned out that not only all the ordinary curves (circle, line segment, ellipse, etc) but also all Cantor curves have dimension one in Urysohn’s sense. Thus, it now became possible to define the notion of a curve in space as well as in the plane.
A curve is a continuum of dimension one.
Taken from Stories about Sets by N.Ya.Vilenkin