Fellow Celts,

JurismathematicoCelt Tom Brennan (tbrennan@math.harvard.edu)
has provided us with this fascinating piece of scientific inquiry:


Attend!  Some Properties of the Celt-space

To begin with, we must note that the Celt-space is not Hausdorff.
That is to say, given two points of the Celt-space it is not in general 
possible to separate them, as elements of the space, using the open 
sets of the topology.  The points in the Celt-space are, of course,
the Celts (the nature of the Celts will be described more fully
below), and the fact that the Space lacks the Hausdorff property is
simply a reflection of the fact that the Celt-space operates to unifty
Celts.  Because the Hausdorff property fails, many of our usual 
intuitions about topological spaces may not give us proper guidance 
in this realm, and so we should be careful about the assumptions we
make.

The next comment that needs to be made about the Celt-space relates
to the limited manner in which we are able to make observations of
it, and thereby to get information about its overall structure.  You
see, the Celt-space intersects our physical world (spacetime or what have 
you), but it is not contained within it.  When observing the Celt-space
in our world, we thus see only a cross-section of the Space and we see
it only as it moves through the physical world.  (Cf. the concept of
the world sheet in physics:  the path etched out by a superstring as it
moves through a multidimensional model of spacetime.)  We are therefore
limited in how fully we are able to describe the overall structure of
the Celt Space, and some aspects of it will necessarily remain a 
mystery.

The points of the Celt-space are known as the Celts.  The relationship
between these Celts and the Celts of common parlance deserves to be
explicated briefly.  Each Celt, in the everyday sense, is the intersection
with our world of a subset of points in the Celt-space which are
identified under a certain natural equivalence relation on the Space.  
Thus, wherever there is a Celt (in the everyday sense) in our world, there
is a point of intersection between our world and the Celt-space.  When the
meaning is clear from context, we will sometimes refer alternatively to
Celts as being either the individual points of the Celt-space or as being
the equivalence classes just described.

The Celt-space is topologically connected, however its intersections with
our world frequently have many connected components.  Often, each
connected component may not have overlap with more than one Celt (in the
equivalence class sense).  However, at times of Celtic Nexus, such a
connected component has overlap with multiple Celts.  At a Celtic Nexus,
it has been observed that the fundamental group of the space resulting
from the local intersection is always finite and cyclic.  In fact, its
order is equal to the number of (equivalence classes) of Celts present at
the Nexus, and the cyclic nature of the group is manifested in the Wheel
of Celticity present at every such Nexus.  Also interesting is the fact
that each local intersection of this sort results in an orientable space,
as is manifested by the fact that the Wheel of Celticity has a naturally
preferred clockwise orientation.

The final result of this note pertains to the fundamental group of the
entire Celt-space:  it is isomorphic to S^1, the circle group.  There are
at least two important things to say about this.  First, this is a further
manifestation of the Wheel of Celticity.  In fact, it is believed that
there is a Universal Wheel of Celticity accompanying this circle group and
that the Wheel of Celticity present at each Celtic Nexus is simply the
intersection of this Universal Wheel with our world at the point of the
Nexus.  Second, the fact that the fundamental group has infintely many
generators means that the Celt-space is not embeddable in any finite
dimensional space. Thus not only has the Celt-space never been known to be
contained completely within our physical world, but it is in fact not
possible for it to do so (in any finite dimensional model of our world).

With further research, more results are undoubtedly obtainable along these
lines.  The best method, at the moment, for deducing results about the
Celt-space at large, seems to be to understand it first locally (in the
Celtic Nexi) and then extrapolate from there.  It is believed that
techniques in this direction should enable one eventually to understand
not only the fundamental group, as described herein, but also the higher
homotopy groups of the space.  In another direction, more creative
techniques need to be developed to understand the nature of the
equivalence relation, mentioned above, which exists naturally on the
Celt-space.  For now though, these and other findings must await future
study.

  -- Tom Brennan, Badass