On Ramsey properties, function spaces, and topological games

An open question of Gruenhage asks if all strategically selectively separable spaces are Markov selectively separable, a game-theoretic statement known to hold for countable spaces. As a corollary of a result by Berner and Juh?sz, we note that the ?strong? version of this statement, where the second player is restricted to selecting single points rather than finite subsets, holds for all T3 spaces without isolated points. Continuing this investigation, we also consider games related to selective sequential separability, and demonstrate results analogous to those for selective separability. In particular, strong selective sequential separability in the presence of the Ramsey property may be reduced to a weaker condition on a countable sequentially dense subset. Additionally, ?- and ?-covering properties on X are shown to be equivalent to corresponding sequential properties on Cp(X). A strengthening of the Ramsey property is also introduced, which is still equivalent to ?2 and ?4 in the context of Cp(X).

Banach Spaces of Almost Universal Complemented Disposition

We introduce and study the notion of space of almost universal complemented disposition (a.u.c.d.) as a generalization of Kadec space. We show that every Banach space with separable dual is isometrically contained as a $1$-complemented subspace of a separable a.u.c.d. space and that all a.u.c.d. spaces with $1$-finite dimensional decomposition (FDD) are isometric and contain isometric $1$-complemented copies of every separable Banach space with $1$-FDD. We then study spaces of universal complemented disposition (u.c.d.) and provide different constructions for such spaces. We also consider spaces of u.c.d. with respect to separable spaces.

Children’s drawings as part of School Language Profiles: Heteroglossic realities in families and schools

Child language development occurs in a given environment, complete with explicit and implicit regulations, intervening actors with their coherent or contradictory intentions, and specific resources for speakers. The main research question of this contribution is: What can drawings as parts of School Language Profiles tell us about the multilingual environments of bilingual families and schools? The analytical framework of spatial and language practices provides a means to talk about how and why certain expressions are chosen and which influences are mentioned in relation to school and family. In particular, the focus is on how heteroglossic spaces are constructed through local/spatial/language practices and how these constructions are represented in the drawings of children. The drawings were collected in a bilingual school in Austria, with Slovene and German as languages of instruction. Children’s drawings present a fine-grained perception of their multilingual surroundings and we see how children refer to home/school in their drawings and distinguish language realities. Findings indicate that language regimes and goals of families and school are in close relation to each other, have influence on each other but do not necessarily always complement each other. This means that in analyzing heteroglossic realities, both cannot be regarded as separate (or separable) spaces.

The unions of dense metrizable subspaces with certain local properties

Many important examples of topological spaces can be represented as a union of a finite or countable collection of metrizable subspaces. However, it is far from clear which spaces in general can be obtained in this way. Especially interesting is the case when the subspaces are dense in the union. We present below several results in this direction. In particular, we show that if a Tychonoff space X is the union of a countable family of dense metrizable locally compact subspaces, then X itself is metrizable and locally compact. We also prove a similar result for metrizable locally separable spaces. Notice in this connection that the union of two dense metrizable subspaces needn?t be metrizable. Indeed, this is witnessed by a well-known space constructed by R.W. Heath.