On a Ramsey-type topological theorem
by
J. Gerlits
Coauthors: Z. Szentmiklóssy

A topological space is left separated if there is a well-order \prec on X and a nbd U(x) for each x in X such that minU(x) = x  (x in X).

Theorem If X is regular and left-separated, then we can colour the pairs of X with 2 colours such that any homogenous set is discrete.

A certain converse was proved recently by S.Todor\'cevi\'c and W.Weiss:

Theorem (S.Todor\'cevi\'c-W.Weiss) If the pairs of a metric space are coloured with n in \omega colours then there exists a not discrete homogenous subset.

With the notation of the partition calculus, this can be expressed as
If X is a not left separated metric space then for any n in \omega  X --> (\omega+1)_n

We extended this result for first countable monotonically normal spaces. (A T₁-space is monotonically normal if to each pair (x, U), where x in X and U is an open nbd of x, we can assign an open set U' such that x in U' and for any two pairs (x, U), (y, V), if x not in V and y not in U then U' \cap V' = \emptyset. Any metrizable and any ordered space is monotonically normal.)

The following property is connected with this subject:

A space X is said to be weakly separated if there exist a nbd-assignment U(x) (x in X) such that any infinite A subset X contains a sequence { x_n: n in \omega} with x_n not in U(x_m) for n < m < \omega.

Theorem If X is regular and weakly left separated, then we can colour the pairs of X with 3(!) colours such that any homogenous set is discrete.

Theorem A monotonically normal space X with countable tightness is weakly left separated iff it is left separated in type |X|.

S.Todor\'cevi\'c-W.Weiss Partitioning Metric Spaces manuscript, September 1995, 1-9

Date received: June 24, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caah-23.