Sequential convergence in topological spaces

In this survey, my aim has been to discuss the use of sequences and countable sets in general topology. In this way I have been led to consider five different classes of topological spaces: first countable spaces, sequential spaces, Frechet spaces, spaces of countable tightness and perfect spaces. We are going to look at how these classes are related, and how well the various properties behave under certain operations, such as taking subspaces, products, and images under proper mappings. Where they are not well behaved we take the opportunity to consider some relevant examples, which are often of special interest. For instance, we examine an example of a Frechet space with unique sequential limits that is not Hausdorff. I asked the question of whether there exists in ZFC an example of a perfectly normal space that does not have countable tightness: such an example was supplied and appears below. In our discussion we shall report two independence theorems, one of which forms the solution to the Moore-Mrowka problem. The results that we prove below include characterisation theorems of sequential spaces and Frechet spaces in terms of appropriate classes of continuous mappings, and the theorem that every perfectly regular countably compact space has countable tightness.Comment: 29 pages. This version incorporates the correction of Proposition 3.2 to include an additional assumption (Hausdorff), whose necessity has been pointed out by Alexander Gouberma

Forbidden rectangles in compacta

We establish negative results about "rectangular" local bases in compacta. For example, there is no compactum where all points have local bases of cofinal type \omega x \omega_2. For another, the compactum \beta\omega has no nontrivially rectangular local bases, and the same is consistently true of \beta\omega \ \omega: no local base in \beta\omega has cofinal type \kappa x c if \kappa < m_{\sigma-n-linked} for some n in [1,\omega). Also, CH implies that every local base in \beta\omega \ \omega has the same cofinal type as one in \beta\omega. We also answer a question of Dobrinen and Todorcevic about cofinal types of ultrafilters: the Fubini square of a filter on \omega always has the same cofinal type as its Fubini cube. Moreover, the Fubini product of nonprincipal P-filters on \omega is commutative modulo cofinal equivalence.Comment: 15 page

On the metrizability of spaces with a sharp base

A base $\mathcal{B}$ for a space $X$ is said to be sharp if, whenever $x\in X$ and $(B_n)_{n\in\omega}$ is a sequence of pairwise distinct elements of $\mathcal{B}$ each containing $x$, the collection $\{\bigcap_{j\le n}B_j:n\in\omega\}$ is a local base at $x$. We answer questions raised by Alleche et al. and Arhangel$'$ski\u{\i} et al. by showing that a pseudocompact Tychonoff space with a sharp base need not be metrizable and that the product of a space with a sharp base and $[0,1]$ need not have a sharp base. We prove various metrization theorems and provide a characterization along the lines of Ponomarev's for point countable bases.Comment: 10 pages. Reprinted from Topology and its Applications, in press, Chris Good, Robin W. Knight and Abdul M. Mohamad, On the metrizability of spaces with a sharp bas

Angelite: Paziteli na vhoda

(bugarski) Statijata prosledjava ikonografskata evoljucija na izobraženijata na arhangelite Mihail i Gavriil, pomesteni pri vhoda na pravoslavnija hram. Povraten moment v neja e 13 vek. Togava arhangel Mihail započva da se izobrazjava kato voin. S tova apotropejnite mu funkcii namirat adekvaten vizualen izraz. K'm kraja na stoletieto arhangel Gavriil započva da se izobrazjava kato pisar - ikonografija, kojato šče b'de dorazvita i utv'rdena prez 14 vek. Prez postvizantijskata epoha v obraza na Mihail se pojavjavat elementi, koito akcentirat v'rhu roljata mu na psihopomp