6 research outputs found
Cardinalities of topologies with small base
Let T be the family of open subsets of a topological space (not necessarily
Hausdorff or even T_0). We prove that if T has a base of cardinality <= mu,
lambda <= mu < 2^lambda, lambda strong limit of cofinality aleph_0, then T has
cardinality = 2^lambda. This is our main conclusion. First we prove
it under some set theoretic assumption, which is clear when lambda = mu ; then
we eliminate the assumption by a theorem on pcf from [Sh 460] motivated
originally by this. Next we prove that the simplest examples are the basic
ones; they occur in every example (for lambda = aleph_0 this fulfill a promise
from [Sh 454]). The main result for the case lambda = aleph_0 was proved in [Sh
454]
Many partition relations below density
We force 2^lambda to be large and for many pairs in the interval
(lambda,2^lambda) a stronger version of the polarized partition relations hold.
We apply this to problems in general topology. E.g. consistently, every
2^lambda is successor of singular and for every Hausdorff regular space X,
hd(X) <= s(X)^{+3}, hL(X) <= s(X)^{+3} and better for s(X) regular, via a
half-graph partition relation. For the case s(X)=aleph_0 we get hd(X), hL(X)
<=aleph_2
On what I do not understand (and have something to say): Part I
This is a non-standard paper, containing some problems in set theory I have
in various degrees been interested in. Sometimes with a discussion on what I
have to say; sometimes, of what makes them interesting to me, sometimes the
problems are presented with a discussion of how I have tried to solve them, and
sometimes with failed tries, anecdote and opinion. So the discussion is quite
personal, in other words, egocentric and somewhat accidental. As we discuss
many problems, history and side references are erratic, usually kept at a
minimum (``see ... '' means: see the references there and possibly the paper
itself).
The base were lectures in Rutgers Fall'97 and reflect my knowledge then. The
other half, concentrating on model theory, will subsequently appear
Analytical Guide and updates for "Cardinal Arithmetic"
Part A: A revised version of the guide in "Cardinal Arithmetic" ([Sh:g]),
with corrections and expanded to include later works.
Part B: Corrections to [Sh:g].
Part C: Contains some revised proof and improved theorems.
Part D: Contains a list of relevant references