Consider a connected T_{1}space
X. Take the Cartesian product of X with itself n times (n ≧ 2) and then remove the
generalized diagonal GD_{n} = {(x_{1},⋯,x_{n}) ∈ X^{n}x_{i} = x_{j} for some i≠j} thus
obtaining the deleted product Z = X^{n} − GD_{n}. If Z should be disconnected
then a great deal can be said about X. For example, if X is compact and
metrizable, then X is homeomorphic to the closed interval [0,1] or to the circle
C = {(x,y) ∈ R^{2}x^{2} + y^{2} = 1}. On the other hand, if it is only assumed (beyond X
being T_{1} and connected and Z disconnected) that X is Hausdorff, locally connected
and separable, then X must be homeomorphic to either (0,1),(0,1],[0,1] or C. In
general, without any assumptions beyond X being T_{1} and connected and Z
disconnected it is possible to define an order on X which is a total order when
restricted to X − a certain finite set, and such that the order topology is coarser
(weaker, smaller) then the original topology on X. Furthermore, all connected
subsets of X and the components of X^{m} − GD_{m} for all m ≧ 2( m not necessarily
equal to n) are determined. In particular the number of components of X^{m} −GD_{m} is
either (m − 1)! or m!∕N!M! where 0 ≦ N,M < n, N + M ≦ m and each of these
numbers is taken on for some X satisfying our hypothesis. The “generalized”
cut point behavior of X is completely determined and an interesting result
is that either there are no cut points or all but at most n points are cut
points.
