Topology
Dimension theory is part of general topology. Other concepts of dimension
can be defined independently in terms of abstract mathematical spaces. A major
task is to show how they relate to each other. Leading figures in many branches
of mathematics have delved into the meaning of dimension including Henri
Lebesgue, L.E.J. Brouwer, Karl Menger, Paul Urysohn and Leopold Vietoris (until
recently the oldest person in Austria, who died in 2002 aged 110).
The pivotal book on the subject was Dimension Theory. Published in 1948 by
Witold Hurewicz and Henry Wallman – it is still seen as a watershed in our
understanding of the concept of dimension.
Dimension in all its forms
From the three dimensions introduced by the Greeks the concept of dimension
has been critically analysed and extended.
The n dimensions of mathematical space were introduced quite painlessly,
while physicists have based theories on space–time (of dimension four) and
recent versions of string theory (see page 97) which demand, 10, 11 and 26
dimensions. There have been forays into fractional dimensions with fractal
shapes (see page 100) with several different measures being studied. Hilbert
introduced an infinite-dimensional mathematical space that is now a basic
framework for pure mathematicians. Dimension is so much more than the one,
two, three of Euclidean geometry.