Products of Michael spaces and completely metrizable spaces

For disjoint subsets of the Michael space has the topology obtained by isolating the points in and letting the points in retain the neighborhoods inherited from . We study normality of the product of Michael spaces with complete metric spaces. There is a ZFC example of a Lindelöf Michael space , of minimal weight , with Lindelöf but with not normal. ( denotes the countable product of a discrete space of cardinality .) If denotes , the normality of implies the normality of for any complete metric space (of arbitrary weight). However, the statement `` normal implies normal' is axiom sensitive.

     

SOME GENERALIZATIONS OF KY FAN''''S BEST APPROXIMATION THEOREM

We pressent new Ky Fan type best approximation theorems for a discontinuous multivalued map on metrizable topological vector spaces and hyperconvex spaces. In addition, fixed point results are derived for the map studied. Our work generalizes severl results in approximation theory.     

Metric spaces and the axiom of choice

We study conditions for a topological space to be metrizable, properties of metrizable spaces, and the role the axiom of choice plays in these matters.     

Two Notes on Top ological Groups

In this short paper, we firstly give a short proof of Birkhoff-Kakutani Theorem by Moore metrizable Theorem. Then we prove that G is a topological group if it is a paratopological group which is a dense G_δ-set in a locally feebly compact regular space X.     

Equicontinuous Seton Locally Convex Spaces

Ｌｅｔ（Ｅ，γ）ｂｅａｌｏｃａｌｌｙｃｏｎｖｅｘｓｐａｃｅａｎｄＥ′ｉｔｓｃｏｎｊｕｇａｔｅｓｐａｃｅ．ＡＥ′ｂｅａｎｅｑｕｉｃｏｎｔｉｎｕ－ｏｕｓｓｅｔｏｎ（Ｅ，γ）．ＴｈｅｗｅｌｌｋｎｏｗｎＡｌａｏｇｌｕ－ＢｏｕｒｂａｋｉＴｈｅｏｒｅｍ（［１］Ｐ２４８）ｓｔａｔｅｓｔｈａｔｅａｃｈｅ－ｑｕｉｃｏｎｔｉｎｕｏｕｓｓｅｔｏｎ（Ｅ，γ）ｉｓσ（Ｅ′，Ｅ）ｒｅｌａｔｉｖｅｌｙｃｏｍｐａｃｔｓｕｂｓｅｔ．Ｎｅｖｅｒｔｈｅｌｅｓｓ，ｅｑｕｉｃｏｎｔｉｎｕｏｕｓｓｅｔｉｓσ（Ｅ′，Ｅ）ｒｅｌａｔｉｖｅｌ…     

Countable compact Hausdorff spaces need not be metrizable in ZF

We show that the existence of a countable, first countable, zero-dimensional, compact Hausdorff space which is not second countable, hence not metrizable, is consistent with ZF.

     

Topological groups in which each nowhere dense subset is closed

Assuming the validity of the combinatorial principlep=c, which follows from Martin's axiom, it is proved that an arbitrary nondiscrete metrizable group topology on an Abelian group can be strengthened to a nondiscrete group topology in which each nowhere dense subset is closed.Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 207–211, August, 1998.     

Densely hereditarily hypercyclic sequences and large hypercyclic manifolds

We prove in this paper that if is a hereditarily hypercyclic sequence of continuous linear mappings between two topological vector spaces and , where is metrizable, then there is an infinite-dimensional linear submanifold of such that each non-zero vector of is hypercyclic for . If, in addition, is metrizable and separable and is densely hereditarily hypercyclic, then can be chosen dense.

     

The Dual Group of a Dense Subgroup

Throughout this abstract, G is a topological Abelian group and $\hat G$ is the space of continuous homomorphisms from G into the circle group ${\mathbb{T}}$ in the compact-open topology. A dense subgroup D of G is said to determine G if the (necessarily continuous) surjective isomorphism $\hat G \to \hat D$ given by $h \mapsto h\left| D \right.$ is a homeomorphism, and G is determined if each dense subgroup of G determines G. The principal result in this area, obtained independently by L. Außenhofer and M. J. Chasco, is the following: Every metrizable group is determined. The authors offer several related results, including these. 1. There are (many) nonmetrizable, noncompact, determined groups. 2. If the dense subgroup D _i determines G _i with G _i compact, then $ \oplus _i D_i $ determines Π_i G _i. In particular, if each G _i is compact then $ \oplus _i G_i $ determines Π_i G _i. 3. Let G be a locally bounded group and let G ⁺ denote G with its Bohr topology. Then G is determined if and only if G ⁺ is determined. 4. Let non $\left( {\mathcal{N}} \right)$ be the least cardinal κ such that some $X \subseteq {\mathbb{T}}$ of cardinality κ has positive outer measure. No compact G with $w\left( G \right) \geqslant non\left( {\mathcal{N}} \right)$ is determined; thus if $\left( {\mathcal{N}} \right) = {\mathfrak{N}}_1 $ (in particular if CH holds), an infinite compact group G is determined if and only if w(G) = ω. Question. Is there in ZFC a cardinal κ such that a compact group G is determined if and only if w(G) < κ? Is $\kappa = non\left( {\mathcal{N}} \right)?\kappa = {\mathfrak{N}}_1 ?$