Locally Constant Functions

Let X be a compact Hausdorff space and M a metric space. E_0(X,M) is the set of f in C(X,M) such that there is a dense set of points x in X with f constant on some neighborhood of x. We describe some general classes of X for which E_0(X,M) is all of C(X,M). These include beta N - N, any nowhere separable LOTS, and any X such that forcing with the open subsets of X does not add reals. In the case that M is a Banach space, we discuss the properties of E_0(X,M) as a normed linear space. We also build three first countable Eberlein compact spaces, F,G,H, with various E_0 properties: For all metric M: E_0(F,M) contains only the constant functions, and E_0(G,M) = C(G,M). If M is the Hilbert cube or any infinite dimensional Banach space, E_0(H,M) is not all of C(H,M), but E_0(H,M) = C(H,M) whenever M is a subset of RR^n for some finite n

Limits in Function Spaces and Compact Groups

If B is an infinite subset of omega and X is a topological group, let C^X_B be the set of all x in X such that converges to 1. If F is a filter of infinite sets, let D^X_F be the union of all the C^X_B for B in F. The C^X_B and D^X_F are subgroups of X when X is abelian. In the circle group T, it is known that C^X_B always has measure 0. We show that there is a filter F such that D^T_F has measure 0 but is not contained in any C^X_B. There is another filter G such that D^X_G = T. We also describe the relationship between D^T_F and the D^X_F for arbitrary compact groups X.Comment: 16 page

Limits in compact abelian groups

Let X be compact abelian group and G its dual (a discrete group). If B is an infinite subset of G, let C_B be the set of all x in X such that <phi(x) : phi \in B> converges to 1. If F is a free filter on G, let D_F be the union of all the C_B for B in F. The sets C_B and D_F are subgroups of X. C_B always has Haar measure 0, while the measure of D_F depends on F. We show that there is a filter F such that D_F has measure 0 but is not contained in any C_B. This generalizes previous results for the special case where X is the circle group.Comment: 14 page

The Deep Origins of Kashmir Shawls, Their Broad Dissemination and Changing Meaning. Or Unraveling the Origins and History of a Unique Cashmere Shawl

Emulation is constant in all forms of art. Debates have arisen regarding the nature of this imitation by Europeans of indigenous Kashmir shawls. The intrinsic Kashmiri aspect was the weave itself: nowhere else was a double interlock tapestry twill technique used. The unique fabric originated in Tibet: pashmina from the underbelly of the mountain goat. The shawl was strong, lightweight, and warm. The earliest Kashmir shawls were simple in design: the double long shawls and moon shawls. The earliest shawls had simple motifs, single floral blooms. By the end of the eighteenth century, this motif was compounded to many blooms or paisley, multiplied across the borders. Europeans discovered these simple shawls and transported them to Europe. They no longer warmed adult men, but embellished women. New local customs arose around the shawls for dowry, christenings. Local drawloom weavers in England and France replicated the shawls, amplifying the colors and design to fit European norms while embracing the singular dominant form, the paisley. The Kashmir shawls in India had a rapid stylist development throughout the nineteenth century. Kashmir shawls maintained their prestige locally and abroad, and after 1840 European merchants requested shawl patterns from Kashmir. In France, manufacturers constructed new looms to closely replicated the design of Kashmir shawls and tapestry weave. French weavers made jacquard imitations with embroidery of mid-century Kashmir pieced shawls. The weave structure reveals the origin of a shawl: if it has wefts that run the horizontal length it is European; if it is non-linear on the back it is Kashmiri. My goal is to demonstrate the deep local nature of Kashmir shawls, their stylistic progression which has been argued to be European in influence but is mainly not, and the complex symbiosis of Kashmir and European production, in the unique battant brocheur weave

Kashmir Shawls: The Perfect Exemplar of a Textile Shaping and Being Shaped

The history of the Kashmir shawl and its stylistic progress and appropriation by other cultures reveals that an art form can be viewed as is (sui generis), and also in a context of social life that is not always what we would like to celebrate. And because the shawls have been made over centuries, this context has changed frequently.1 The Kashmir shawl is very special because it combines: a material that is extremely rare and luxurious which we now call cashmere or pashmina, a weave structure that is unique in its region and very complex, a symbolism of design motifs that is still debated, color dyes still not fully explored, a design that is also unique for its region. (All the shawls illustrated in this discussion are from my collection and cannot be reproduced.) It is not possible to provide in-depth coverage of all the peregrinations, changes and influences on shawl creation. I will focus on the eighteenth- to the late nineteenth century developments

Spaces with No S or L Subspaces

We show it consistent for spaces X and Y to be both HS and HL even though their product X ×Y contains an S-space. Recall that an S-space is a T3 space that is HS but not HL. More generally, consider spaces that contain neither an S-space nor an L-space. We say a space is ESLC iff each of its subspaces is either both HS and HL or neither HS nor HL. The C in ESLC refers to HC; a space is HC iff each of its subspaces has the ccc (countable chain condition) (iff the space has no uncountable discrete subspaces). Classes of ESLC spaces include metric spaces (because every metric space is either second countable or has an uncountable discrete subspace), subspaces of the Sorgenfrey line and (suitably defined) generalized butterfly spaces; for these classes, countable products are still ESLC