Locally compact, ω1\omega_1-compact spaces

An $\omega_1$-compact space is a space in which every closed discrete subspace is countable. We give various general conditions under which a locally compact, $\omega_1$-compact space is $\sigma$-countably compact, i.e., the union of countably many countably compact spaces. These conditions involve very elementary properties.Comment: 21 pages, submitted, comments are welcom

A Compact Homogeneous S-space

Under the continuum hypothesis, there is a compact homogeneous strong S-space.Comment: 6 page

Thermofield-Bosonization on Compact Space

We develop the construction of fermionic fields in terms of bosonic ones to describe free and interaction models in the circle, using thermofielddynamics. The description in the case of finite temperature is developed for both normal modes and zero modes. The treatment extends the thermofield-bosonization for periodic space

On the weak and pointwise topologies in function spaces

For a compact space $K$ we denote by $C_w(K)$ ($C_p(K)$) the space of continuous real-valued functions on $K$ endowed with the weak (pointwise) topology. In this paper we address the following basic question which seems to be open: Suppose that $K$ is an infinite (metrizable) compact space. Is it true that $C_w(K)$ and $C_p(K)$ are homeomorphic? We show that the answer is "no", provided $K$ is an infinite compact metrizable $C$-space. In particular our proof works for any infinite compact metrizable finite-dimemsional space $K$

Homogeneous Subspaces of Products of Extremally Disconnected Spaces

Homogeneous countably compact spaces $X$ and $Y$ whose product $X\times Y$ is not pseudocompact are constructed. It is proved that all compact subsets of homogeneous subspaces of the third power of an extremally disconnected space are finite. Moreover, under CH, all compact subsets of homogeneous subspaces of any finite power of an extremally disconnected space are finite and all compact subsets of homogeneous subspaces of the countable power of an extremally disconnected space are metrizable. It is also proved that all compact homogeneous subspaces of finite powers of an extremally disconnected space are finite, which strengthens Frol\'{\i}k's theorem

Compact composition operators on the Dirichlet space and capacity of sets of contact points

In this paper, we prove that for every compact set of the unit disk of logarithmic capacity 0, there exists a Schur function both in the disk algebra and in the Dirichlet space such that the associated composition operator is in all Schatten classes (of the Dirichlet space), and for which the set of points whose image touches the unit circle is equal to this compact set. We show that for every bounded composition operator on the Dirichlet space and for every point of the unit circle, the logarithmic capacity of the set of point having this point as image is 0. We show that every compact composition operator on the Dirichlet space is compact on the gaussian Hardy-Orlicz space; in particular, it is in every Schatten class on the usual Hilbertian Hardy space. On the other hand, there exists a Schur function such that the associated composition operator is compact on the gaussian Hardy-Orlicz space, but which is not even bounded on the Dirichlet space. We prove that the Schatten classes on the Dirichlet space can be separated by composition operators. Also, there exists a Schur function such that the associated composition operator is compact on the Dirichlet space, but in no Schatten class