## J. Gerlits, Z. Szentmiklóssy

## On a Ramsey-type topological theorem

A topological space is * left separated * if there is a
well-order < on ` X ` and a nbd `U(x)` for each
`x` in `X` such
that ` min U(x)=x ` for each ` 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
homogeneous set is discreet.

A certain converse was proved recently by S.Todorcevic and W.Weiss:

** Theorem (S.Todorcevic-W.Weiss) ** * If the
*`r`-tuples of a not left separated metric space are coloured with
` n\in omega `colours then
there exists a not discreet homogeneous 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 `2 <= r, n < omega` we have `X --> (omega +1)_n^r`

We extended this result for first countable monotonically normal
spaces. (A `T_1`-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 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 homogeneous set is discreet.

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

** Reference **

S.Todorcevic--W.Weiss: Partitioning Metric Spaces, manuscript, September 1995, 1-9.

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