cess was to turn the collected data- sets into meaning- fi lled points, lines,
and planes, i.e. into geometrical entities which could be projected onto a
sheet of papyrus, the cartographer’s version of the painter’s canvas and
the movie- maker’s screen. This third step of projection was in many ways
the most crucial, for it was through this operation that the abstract con-
cepts of lines and labels were ontologically transformed into material ob-
jects—the points turned into cities, small islands, and the mouths of minor
rivers; the lines becoming
The primary tool for the study of the structure of topological spaces and their
relationship with each other is the notion of a continuous mapping.
Let X and Y be topological spaces, and let X -+ Y be a mapping of sets.
This rp is said to be a continuous mapping if, for each open 0 in Y, rp-110) is
open in X That. is, a mapping is continuous if inverse images of open sets are
open. We shall be concerned for the remainder of this chapter with under-
standing this definition.
Example. Let X and Y be topological spaces, with X
Subsets and Mappings
It is convenient at this point to introduce some definitions and notation, and
to recall a few facts about sets, subsets, and mappings. It is, in my opinion, a
waste of time to try to memorize all the various properties listed below.
Rather, one should try to become skillful at guessing with reasonable accuracy
what is true, and to become adept at quickly finding the (always easy) proofs
and at effortlessly manipulating the symbols.
Fix a set S. We denote by 0, the empty set, the subset of S having no
elements. For A and B subsets
. And mathematics provides examples of the general map anal-
ogy. (Causal maps— so important to the life and social sciences— will be
addressed in the next chapter.)
[ 150 ] Chapter Si x
Extreme- Scale Maps in Cosmology
The largest thing humans can map is our universe. There are paradoxes
and qualifications involved in mapping the biggest existence in space and
time, with the fastest, yet finite, travel velocity that we know of: light.
For instance, we cannot measure and know what lies outside the limits of
the universe that we see with incoming
Exploring, Traveling, Mapping
Part 1 established some of the ways in which the Enlightenment can be
thought of geographically—thought of, that is, in terms of its national ex-
pression, its international and cosmopolitan making, and its reception in
local sites and social spaces. Attention was drawn both to the limited utility
of the Enlightenment and to the Enlightenment as a national phenomenon,
and to the diff erences that questions of place and scale and movement make
to a richer understanding of the Enlightenment’s where.
Part 2 is concerned with
self- educated person gains from independent
reading and clings to in defiance of what anyone else tries to tell him” (E. S.
Morgan 2009, 6).
2. McLean 1992; Sardar 1992; Mann 2011.
[ 178 ] Chapter Seven
the fervor of his confidence in his calculations. You see, Columbus had
prepared heavily for his round of investor pitches. Having studied Pliny,
Ptolemy, and Marco Polo, among many others, and corresponded with
the Florentine astronomer, astrologer, geographer, and cartographer
Paolo dal Pozzo Toscanelli, Columbus optimistically calculated