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In article <1992Nov5.004804.24757@galois.mit.edu> jbaez@riesz.mit.edu
(John C. Baez) writes:

>Anyway, manifolds come in several flavors, most notably "smooth" and
>"topological".  A topological manifold is a Hausdorff space such that every
>point has a neighborhood homeomorphic to R^n (Euclidean n-space).

A bit of history may be helpful.  Most of us, even those without much 
background in abstract mathematics, are familiar with the idea that a
surface in R^3 can be defined by an equation of the form z = f(x,y) or
f(x,y,z) = 0.  This suggests a natural generalization to higher dimensions:
define a manifold to be the set M of all points in R^n satisfying an
equation of the form f(x_1,...,x_n) = 0.  This, roughly, is how manifolds
were first defined.  If f is smooth (infinitely differentiable), then M
is called a smooth manifold; if f is a polynomial, then M is called an
algebraic variety; and so on.

Well, this definition has the advantage of being easily motivated and
concrete, but as people started studying manifolds more deeply they
found that it was often a nuisance to have to be tied to a particular
function f to define their manifolds.  There were certain geometric
properties of manifolds that were "intrinsic" to the surface and didn't
really depend on the function f in any essential way.  It seemed that
what was needed was a way of defining manifolds without having to pick
a space R^n for the manifold to live in and specify an explicit function.

Hence the modern approach to manifolds is to define them as objects in
their own right, without reference to a space that they're imbedded
in.  This is what motivates the definition that John Baez gives.
Notice that he doesn't require the manifold to live in some R^n.  The
extra abstraction is a small price to pay for the simplification and
logical clarity that it yields, as you will appreciate if you study the
subject more deeply.  Similarly, smooth manifolds and algebraic
varieties (and later schemes) are nowadays defined without making them
live in R^n.
-- 
Tim Chow     tycchow@math.mit.edu
Where a calculator on the ENIAC is equipped with 18,000 vacuum tubes and weighs
30 tons, computers in the future may have only 1,000 vacuum tubes and weigh
only 1 1/2 tons.                               ---Popular Mechanics, March 1949