In this paper an Eilenberg-type characterization of unicoherence at subcontinua and a mapping property about this unicoherence are given.

In this paper, we define the notion of a proper helix of order $d$ in a pseudo-Riemannian manifold and investigate those curves in a totally umbilical pseudo-Riemannian submanifold.

It is proved that if $X$ is an $n$-dimensional closed convex subset in a linear metric space $E$, then there is a retraction $r:E\rightarrow X$ such that $\Vert x-r(x)\Vert\leq 2(n+1)\Vert x-X\Vert$ for every $x\in E$. This fact is applied to study the AR-property in linear metric spaces. We identify a class of Roberts spaces with the AR-property. We also give a direct proof that for every $p\in[0,1 )$,$L_{\rho}$ is a needle point space.

We exhibit in this article some classes of spaces for which properties $\gamma$ and $\gamma_{p}$ are countable additive, and we prove that for some type of spaces and ultrafilters $p\in\omega^{*}, \gamma$ is equivalent to $\gamma_{p}$. We obtain: (1) If $\{X_{n}\}_{n \lt \omega}$ is a sequence of metrizable locally compact spaces with $\gamma_{p}(p\in\omega^{*})$, then $\Pi_{n \lt \omega}C_{\pi}(X_{n})$ is a $FU(p)$-space; (2) $C_{\pi}(X)$ is a Fréchet-Urysohn (resp., $FU(p)$) space iff $C_{\pi}(F(X))$ has the same property, where $F(X)$ is the free topological group generated by $X$; (3) For a locally compact metrizable and non countable space $X$, $C_{\pi}(X)$ is a Fréchet-Urysohn (resp., $FU(p)$) space iff $C_{\pi}(L_{\pi}(X))$ is Fréchet-Urysohn (resp., $FU(p)$), where $L_{\pi}(X)$ is the dual space of $C_{\pi}(X)$; (4) For every $\check{C}$ech complete space $X$ and every $p\in\omega^{*}$ for which $R$ does not have $\gamma_{p}$, $C_{\pi}(X)$ is Fréchet-Urysohn iff $C_{\pi}(X)$ is a $FU(p)$-space. Also we give some results concerning $P$-points in $\omega^{*}$ related with $p$-Fréchet-Urysohn property and topological function spaces.

We treat Killing-transversally symmetric spaces (briefly, KTS-spaces), that is, Riemannian manifolds equipped with a complete unit Killing vector field such that the reflections with respect to the flow lines of that field can be extended to global isometries. Such manifolds are homogeneous spaces equipped with a naturally reductive homogeneous structure and they provide a rich set of examples of reflection spaces. We prove that each simply connected reducible KTS-space $M$ is a Riemannian product of a symmetric space $M^{\prime}$ and a special kind of KTS-space $M^{\prime\prime}$, called a contact KTS-space. Such a particular manifold $M^{\prime\prime}$ is an irreducible, odd-dimensional principal $G^{1}$-bundle over a Hermitian symmetric space. The main purpose of the paper is to give a classification of this special class of manifolds $M^{\prime\prime}$.

Resolutions of spaces can be viewed as special inverse systems, which behave very much like inverse systems behave in the compact case. T. Watanabe defined a category of polyhedral resolutions and showed that the limit functor defines a natural equivalence between this category and the category of topologically complete spaces. In order to develop his theory he had to consider gauged inverse systems, i.e., inverse systems whose terms are endowed with certain coverings, called meshes. This paper is devoted to the question if one can develop an analogous theory for usual (nongauged) inverse systems. An example is exhibited, which suggests a negative answer.

If $T\in \mathscr{L}(X)$ then $T$ is analytic if and only if $(\lambda-T)^{-1}$ has a pole for at least a $\lambda\in\sigma(T)$. Furthermore, every analytic operator has a non-trivial invariant subspace.

A characterization of self-dual connections of homogeneous principal bundles over quatemionic Kaehler symmetric spaces in terms of the holonomy homomorphism, is given using the group theoretic approach.

Let $C$ be a complete nonsingular curve over an algebrically closed field $K$ and $L$ a very ample invertible sheaf on $C$. We denote by $\phi_{L}$: $C\rightarrow P(H^{0}(L))$, the projective embedding of $C$ by means of the vector space $H^{0}(C, L)$. There are two purposes in this paper. One is to the question: What is the intersection of quadrics through $\phi_{L}(C)$? The other is to answer the question: What degrees are the minimal generators of the associated homogeneous ideal?

In this paper we introduce a quadric representation $\varphi$ of the product of two pseudo-Riemannian isometric immersions. We characterize the product of submanifolds whose quadric representation satisfies $\Delta H_{\varphi}=\lambda H_{\varphi}$, for a real constant $\lambda$, where $H_{\varphi}$ is the mean curvature vector field of $\varphi$. As for hypersurfaces, we prove that the only ones satisfying that equation are minimal products as well as products of a minimal hypersurface and another one which has constant mean and constant scalar curvatures with an appropriate relation between them. In particular, the family of these surfaces consists of $H^{2}(-1)$ and $S^{1}(2/3)\times H^{1}(-2)$ in $S_{1}^{3}(1)$ and $S_{1}^{2}(1),H_{1}^{1}(-2/3)\times S^{1}(2)$, $S_{1}^{1}(2)\times H^{1}(-2/3)$ and a B-scroll over a null Frenet curve with torsion $\pm\sqrt{2}$ in $H_{1}^{3}(-1)$.

We prove the following theorem which is a locally compact analogue of results of $S$. Ferry and the author. Theorem. Let $f:X\rightarrow Y$ be a proper map between finite dimensional locally compact polyhedra $X$ and Y. Suppose that (1) $\pi_{j}(f):\pi_{i}(X)\rightarrow\pi_{i}(Y)$ is an isomorphism for each $i\leq n$, (2) $f$ induces a surjection between the ends of $X$ and $Y$, and (3) $f$ induces an isomorphism between the $i$-th homotopy groups of ends of $X$ and $Y$ for each $i\leq n$. Then there exist a locally compact polyhedron $Z$ and proper $UV^{n}$-maps $\alpha:Z\rightarrow X$ and $\beta:Z\rightarrow Y$ such that (4) $\dim Z\leq 2\max(\dim X,n)+3$, (5) $f\circ\alpha$ and $\beta$ is properly $n$-homotopic, and (6) $\alpha$ has at most countably many non-contractible fibre all of which have the homotopy type of $S^{n+1}$.

The hyperspace $C(X)$ of a continuum $X$ is always arcwise connected. In [6], S. B. Nadler Jr. and J. Quinn show that if $C(X)-\{A_{j}\}$ is arcwise connected for each $i=1,2$, then $C(X)-\{A_{1},A_{2}\}$ is also arcwise connected. Nadler raised questions in his book [5]: Is it still true with the two sets $A_{1}$ and $A_{2}$ replaced by $n$ sets, $n$ finite? What about countably many? What about a collection $\{A_{\lambda}:\lambda\in\Lambda\}$ which is a compact zero-dimensional subset of the hyperspace? In this paper we prove that if $\mathscr{A}\subset C(X)$ is a closed countable subset, $\mathscr{U}$ is an arc component of an open set of $C(X)$ and $C(X)-\{A\}$ is arcwise connected for each $A\in mathscr{A}$, then $\mathscr{U}-\mathscr{A}$ is arcwise connected.

A module over a Gorenstein ring is said to be Gorenstein injective if it splits under all modules of finite projective dimension. We show that over a Gorenstein ring every module has a Gorenstein injective envelope. We apply this result to the group algebra $\hat{Z}_{\rho}G$ (with $G$ a finite group and $\hat{Z}_{p}$ the ring of $p$-adic integers for some prime $p$) and show that ever finitely generated $\hat{Z}_{p}G$-module has a cover by a lattice. This gives a way of lifting finite dimensional representations of $G$ over $Z/(p)$ to modular representations of $G$ over $\hat{Z}_{p}$.

We give a classification of real hypersurfaces in complex projective space under assumptions that the structure vector $\xi$ is principal, the focal map has constant rank and that $\nabla_{\xi}C=0$, where $C$ is the Weyl conformal curvature tensor of the real hypersurface.