Perfect mappings in topological groups, cross-complementary subsets and quotients

summary:The following general question is considered. Suppose that $G$ is a topological group, and $F$, $M$ are subspaces of $G$ such that $G=MF$. Under these general assumptions, how are the properties of $F$ and $M$ related to the properties of $G$? For example, it is observed that if $M$ is closed metrizable and $F$ is compact, then $G$ is a paracompact $p$-space. Furthermore, if $M$ is closed and first countable, $F$ is a first countable compactum, and $FM=G$, then $G$ is also metrizable. Several other results of this kind are obtained. An extensive use is made of the following old theorem of N. Bourbaki [5]: if $F$ is a compact subset of a topological group $G$, then the natural mapping of the product space $G\times F$ onto $G$, given by the product operation in $G$, is perfect (that is, closed continuous and the fibers are compact). This fact provides a basis for applications of the theory of perfect mappings to topological groups. Bourbaki's result is also generalized to the case of Lindelöf subspaces of topological groups; with this purpose the notion of a $G_\delta$-closed mapping is introduced. This leads to new results on topological groups which are $P$-spaces

The rank of the diagonal and submetrizability

summary:Several topological properties lying between the submetrizability and the $G_\delta$-diagonal property are studied. We are mostly interested in their relationship to each other and to the submetrizability. The first example of a Tychonoff space with a regular $G_\delta$-diagonal but without a zero-set diagonal is given. The same example shows that a Tychonoff separable space with a regular $G_\delta$-diagonal need not be submetrizable. We give a necessary and sufficient condition for submetrizability of a regular separable space. The rank $5$-diagonal plays a crucial role in this criterion. Every closed bounded subset of a Tychonoff space with a $G_\delta$-diagonal is shown to be Čech-complete. Under a slightly stronger condition, any such subset is shown to be a Moore space. We also establish that every closed bounded subset of a Tychonoff space with a regular $G_\delta$-diagonal is metrizable by a complete metric and, therefore, has the Baire property. Some further results are obtained, and new open problems are posed

Addition theorems and DD-spaces

summary:It is proved that if a regular space $X$ is the union of a finite family of metrizable subspaces then $X$ is a $D$-space in the sense of E. van Douwen. It follows that if a regular space $X$ of countable extent is the union of a finite collection of metrizable subspaces then $X$ is Lindelöf. The proofs are based on a principal result of this paper: every space with a point-countable base is a $D$-space. Some other new results on the properties of spaces which are unions of a finite collection of nice subspaces are obtained

A note on condensations of Cp(X)C_p(X) onto compacta

summary:A condensation is a one-to-one continuous mapping onto. It is shown that the space $C_p(X)$ of real-valued continuous functions on $X$ in the topology of pointwise convergence very often cannot be condensed onto a compact Hausdorff space. In particular, this is so for any non-metrizable Eberlein compactum $X$ (Theorem 19). However, there exists a non-metrizable compactum $X$ such that $C_p(X)$ condenses onto a metrizable compactum (Theorem 10). Several curious open problems are formulated

Supergravity Black Holes and Billiards and Liouville integrable structure of dual Borel algebras

In this paper we show that the supergravity equations describing both cosmic billiards and a large class of black-holes are, generically, both Liouville integrable as a consequence of the same universal mechanism. This latter is provided by the Liouville integrable Poissonian structure existing on the dual Borel algebra B_N of the simple Lie algebra A_{N-1}. As a by product we derive the explicit integration algorithm associated with all symmetric spaces U/H^{*} relevant to the description of time-like and space-like p-branes. The most important consequence of our approach is the explicit construction of a complete set of conserved involutive hamiltonians h_{\alpha} that are responsible for integrability and provide a new tool to classify flows and orbits. We believe that these will prove a very important new tool in the analysis of supergravity black holes and billiards.Comment: 48 pages, 7 figures, LaTex; V1: misprints corrected, two references adde

Two types of remainders of topological groups

summary:We prove a Dichotomy Theorem: for each Hausdorff compactification $bG$ of an arbitrary topological group $G$, the remainder $bG\setminus G$ is either pseudocompact or Lindelöf. It follows that if a remainder of a topological group is paracompact or Dieudonne complete, then the remainder is Lindelöf, and the group is a paracompact $p$-space. This answers a question in A.V. Arhangel'skii, {\it Some connections between properties of topological groups and of their remainders\/}, Moscow Univ. Math. Bull. 54:3 (1999), 1--6. It is shown that every Tychonoff space can be embedded as a closed subspace in a pseudocompact remainder of some topological group. We also establish some other results and present some examples and questions

GδG_\delta-modification of compacta and cardinal invariants

summary:Given a space $X$, its $G_\delta$-subsets form a basis of a new space $X_\omega$, called the $G_\delta$-modification of $X$. We study how the assumption that the $G_\delta$-modification $X_\omega$ is homogeneous influences properties of $X$. If $X$ is first countable, then $X_\omega$ is discrete and, hence, homogeneous. Thus, $X_\omega$ is much more often homogeneous than $X$ itself. We prove that if $X$ is a compact Hausdorff space of countable tightness such that the $G_\delta$-modification of $X$ is homogeneous, then the weight $w(X)$ of $X$ does not exceed $2^\omega$ (Theorem 1). We also establish that if a compact Hausdorff space of countable tightness is covered by a family of $G_\delta$-subspaces of the weight $\leq c=2^\omega$, then the weight of $X$ is not greater than $2^\omega$ (Theorem 4). Several other related results are obtained, a few new open questions are formulated. Fedorchuk's hereditarily separable compactum of the cardinality greater than $c=2^\omega$ is shown to be $G_\delta$-homogeneous under CH. Of course, it is not homogeneous when given its own topology

Addition theorems for dense subspaces

summary:We study topological spaces that can be represented as the union of a finite collection of dense metrizable subspaces. The assumption that the subspaces are dense in the union plays a crucial role below. In particular, Example 3.1 shows that a paracompact space $X$ which is the union of two dense metrizable subspaces need not be a $p$-space. However, if a normal space $X$ is the union of a finite family $\mu$ of dense subspaces each of which is metrizable by a complete metric, then $X$ is also metrizable by a complete metric (Theorem 2.6). We also answer a question of M.V. Matveev by proving in the last section that if a Lindelöf space $X$ is the union of a finite family $\mu$ of dense metrizable subspaces, then $X$ is separable and metrizable

On topological and algebraic structure of extremally disconnected semitopological groups

summary:Starting with a very simple proof of Frol'\i k's theorem on homeomorphisms of extremally disconnected spaces, we show how this theorem implies a well known result of Malychin: that every extremally disconnected topological group contains an open and closed subgroup, consisting of elements of order $2$. We also apply Frol'\i k's theorem to obtain some further theorems on the structure of extremally disconnected topological groups and of semitopological groups with continuous inverse. In particular, every Lindelöf extremally disconnected semitopological group with continuous inverse and with square roots is countable, and every extremally disconnected topological field is discrete

On a theorem of W.W. Comfort and K.A. Ross

summary:A well known theorem of W.W. Comfort and K.A. Ross, stating that every pseudocompact group is $C$-embedded in its Weil completion [5] (which is a compact group), is extended to some new classes of topological groups, and the proofs are very transparent, short and elementary (the key role in the proofs belongs to Lemmas 1.1 and 4.1). In particular, we introduce a new notion of canonical uniform tightness of a topological group $G$ and prove that every $G_\delta$-dense subspace $Y$ of a topological group $G$, such that the canonical uniform tightness of $G$ is countable, is $C$-embedded in $G$