A one-day conference in the series Set theory and its neighbours will take place on Wednesday 5th December at the Department of Mathematics, University College London, 25 Gordon Street, London, WC1.
The talks will be in room 706, with the
first talk starting at 2pm. (Note the later starting time than
usual.)
There will be tea (in room 606) from 4-4.30, and the meeting
should finish by approximately 6.40pm.
The speakers at the meeting will be:
The study of MCP inspired us to investigate monotone versions of different
characterizations of countable paracompactness. In particular, monotonizing
Mack's result that a space is countably paracompact iff its product with [0,1]
is δ-normal led us to the notion of monotone δ-normality
(MδN). It turns out that if the product of a space X with [0, 1] is
MδN then X is MCP, however the converse is not true.
Since much is known about MCP, it seems natural to ask about the behaviour of
MδN and other monotone versions of δ-normality. We show how
several monotonizations of δ-normality interrelate and provide
factorizations of monotone normality in terms of these properties.
I will begin the talk with describing the particular breed of T0
space that tends to arise in semantics, called stably compact space, and I
will explain that it relates one-to-one to the much better known concept
of a compact ordered space.
When trying to define Borel measure on a stably compact space, one finds
that it is convenient to begin with the much simpler concept of a
valuation, which provides a "volume" for the open sets only. I will
describe these and show that there is a perfectly good notion of
integration for (semi) continuous functions with respect to a valuation.
The question then arises whether a valuation on a stably compact space
will determine a Borel measure on the associated compact ordered space.
This is indeed the case and the remainder of the talk will be devoted to
a particularly elegant argument for showing this. (It was my co-author
Klaus Keimel who first suggested this approach.)
We aim to keep the meetings fairly relaxed, allowing plenty of
opportunity for informal discussion. We welcome and encourage anyone
to participate. Please do tell anyone about the meeting who you think
may be interested in it. There is no registration fee for the meeting.
We are happy for you to email us to let us know if
you intend to come, but you are also very welcome simply to turn up on
the day if you make a late decision. And let us know if you would
like to speak or have ideas for speakers at future meetings.
After the meeting we will
probably go for a near-by drink and then supper.
Large Cardinals and Definable Well-Orderings of the Universe
Abstract: One very nice property of Goedel's constructible universe L is that it
carries a definable well-ordering of all sets. Of course, many large
cardinals axioms are incompatible with the assumption that V=L, but
nevertheless we can still ask whether they are compatible with the
existence of a definable well-ordering of the universe. We shall exhibit a
forcing construction that demonstrates for many large cardinal axioms that
they are indeed compatible, using a coding scheme involving the principle
◊*.
MCP, MδN and factorizations of MN
Abstract: Ishikawa characterized a countably paracompact space as one in which given any
decreasing sequence of closed sets (Di) with empty intersection, there
exists a sequence of open sets (Ui) whose closures have empty intersection
such that Ui contains Di for each i. A monotone version of Ishikawa's
property, known as monotone countable paracompactness (MCP), was introduced by
Good, Knight and Stares who carried out an extensive survey of it. In many
cases, the set-theoretic assumptions necessary for results concerning countable
paracompactness may be abandoned with MCP. In this way MCP is analogous to
monotone normality; set-theoretic assumptions may be dropped when replacing
'normal' with 'monotonically normal' in certain results.
From valuations to measures
Abstract: There is a long history of the use of topology in computer
science, going back at least to the work of Kleene in the 50s. The
structures considered in Denotational Semantics, in particular, are
best understood as certain topological spaces. Unfortunately, however, the
spaces are just T0 and not Hausdorff and so many of our ingrained
topological intuitions do not apply.
Ramsey's Theorem for graphs under degree conditions
Abstract: tba
Decomposing 2 → 3 and 3 → 3
Abstract: In this talk, I will review game-theoretic
characterizations of the continuous, Baire class 1,
and Baire class 2 functions, and describe
how the Jayne-Rogers theorem can be generalized using
game-theoretic methods.
Let "n → m" denote the class of functions for which the preimage
of a boldface Σ0n set is boldface
Σ0m.
The Jayne-Rogers theorem states (with very minimal assumptions)
that the 2 → 2 functions are piecewise continuous. The results
I describe in this talk are for the 2 → 3 and 3 → 3 classes,
for functions on the Baire space.
Return to the Set theory and its neighbours homepage for information, including slides from the talks and related preprints, about the previous meetings.
Last updated on 15th November 2007, Charles Morgan
