Posted by Stephen Watson on December 15, 1997 at 15:44:13:

In Reply to: a brief definition posted by Melanie Meyer on December 15, 1997 at 15:41:56:

Dear Melanie,

Thanks for your question.

Stephen Barr in his 1964 book "Experiments in Topology" said "topology is
curiously hard to define... (it) started as a kind of geometry but it has
reached into many other mathematical fields." "In one sense it is the
study of continuity: beginning with the continuity of space, or shapes, it
generalizes and then by analogy leads into other kinds of continuity- and
space as we usually understand it is left far behind.... A topologist is
interested in those properties of a thing that while they are in a sense
geometrical are the most permanent- the ones that will survive distortion
and stretching."

But modern topology goes much further than this:

The great topologist Paul Alexandroff said in 1932:

"The development of set-theoretic methods in topology first led to (the)
discovery (of strange phenomena) and consequently, to a substantial
extension of the idea of space. I would formulate the basic problem of
set-theoretic topology as follows: to determine which set-theoretic
structures have a connection with the intuitively given material of
elementary polyhedral topology and hence deserve to be considered as
geometrical figures- even if very general ones." 

I hope this helps a little. May we put your question (and my reply) in
our "Ask a Topologist" feature? 


Professor Stephen Watson

Department of Mathematics
York University
4700 Keele St., 
North York, Ontario M3J1P3
CANADA