Covering dimension and finite-to-one maps

Hurewicz' characterized the dimension of separable metrizable spaces by means of finite-to-one maps. We investigate whether this characterization also holds in the class of compact F-spaces of weight c. Our main result is that, assuming the Continuum Hypothesis, an n-dimensional compact F-space of weight c is the continuous image of a zero-dimensional compact Hausdorff space by an at most 2n-to-1 map

Isometry groups among topological groups

It is shown that a topological group G is topologically isomorphic to the isometry group of a (complete) metric space iff G coincides with its G-delta-closure in the Rajkov completion of G (resp. if G is Rajkov-complete). It is also shown that for every Polish (resp. compact Polish; locally compact Polish) group G there is a complete (resp. proper) metric d on X inducing the topology of X such that G is isomorphic to Iso(X,d) where X = l_2 (resp. X = Q; X = Q\{point} where Q is the Hilbert cube). It is demonstrated that there are a separable Banach space E and a nonzero vector e in E such that G is isomorphic to the group of all (linear) isometries of E which leave the point e fixed. Similar results are proved for an arbitrary complete topological group.Comment: 30 page

Dimension of graphoids of rational vector-functions

Let $F$ be a countable family of rational functions of two variables with real coefficients. Each rational function $f\in F$ can be thought as a continuous function $f:dom(f)\to\bar R$ taking values in the projective line $\bar R=R\cup\{\infty\}$ and defined on a cofinite subset $dom(f)$ of the torus $\bar R^2$. Then the family \F determines a continuous vector-function $F:dom(F)\to\bar R^F$ defined on the dense $G_\delta$-set $dom(F)=\bigcap_{f\in F}dom(F)$ of $\bar R^2$. The closure $\bar\Gamma(F)$ of its graph $\Gamma(F)=\{(x,f(x)):x\in dom(F)\}$ in $\bar R^2\times\bar R^F$ is called the {\em graphoid} of the family $F$. We prove the graphoid $\bar\Gamma(F)$ has topological dimension $dim(\bar\Gamma(F))=2$. If the family $F$ contains all linear fractional transformations $f(x,y)=\frac{x-a}{y-b}$ for $(a,b)\in Q^2$, then the graphoid $\bar\Gamma(F)$ has cohomological dimension $dim_G(\bar\Gamma(F))=1$ for any non-trivial 2-divisible abelian group $G$. Hence the space $\bar\Gamma(F)$ is a natural example of a compact space that is not dimensionally full-valued and by this property resembles the famous Pontryagin surface.Comment: 20 page

The structure of a the C*-algebra of a locally injective surjection

We obtain a description of the C*-algebras which can occur as a simple quotient of the C*-algebra of a locally injective surjection on a compact metric space of finite covering dimension

Wokół pierwszej polskiej systematyki zamówień. O zapomnianej propozycji Józefa Obrębskiego

The author discusses the first Polish systematics of magic spells against illness, which was proposed by Józef Obrębski in a brochure Index for „Treating the Polish people” by Henryk Biegeleisen in 1931. Obrębski introduced a division of spells into 9 groups, separated on the basis  of their  content,  form and  function.  It  was  not  an  autonomous  development.  It  have been created for the needs of someone else’s work and on the margin of a broader classification of material in the field of folk medicine as a whole. However, the proposal of Obrębski  can  be  regarded  as  a  serious  outline covering  all  material  concerning  the  systematics  of spells against illness. In the following parts of the article the author sets the discussed work in  a  due  context,  locating  it  in  the  history  of Polish research  on  the  genre  of  spells.  She recalls the first nineteenth-century works and analyses a possible impact of the Obrębski’s systematics on the subsequent approaches to the subject.The author discusses the first Polish systematics of magic spells against illness, which was proposed by Józef Obrębski in a brochure Index for „Treating the Polish people” by Henryk Biegeleisen in 1931. Obrębski introduced a division of spells into 9 groups, separated on the basis  of their  content,  form and  function.  It  was  not  an  autonomous  development.  It  have been created for the needs of someone else’s work and on the margin of a broader classification of material in the field of folk medicine as a whole. However, the proposal of Obrębski  can  be  regarded  as  a  serious  outline covering  all  material  concerning  the  systematics  of spells against illness. In the following parts of the article the author sets the discussed work in  a  due  context,  locating  it  in  the  history  of Polish research  on  the  genre  of  spells.  She recalls the first nineteenth-century works and analyses a possible impact of the Obrębski’s systematics on the subsequent approaches to the subject

A note on an example by van Mill

Improving on an earlier example by J. van Mill, we prove that there exists a zero-dimensional compact space of countable pi-weight and uncountable character which is homogeneous under MA+notCH, but not under CH.Comment: Revision after referee's comment

Cosmic dimensions

Martin's Axiom for $\sigma$-centered partial orders implies that there is a cosmic space with non-coinciding dimensions.Comment: 2005-11-15: new version, largely rewritte

On pseudo-k-spaces

[EN] In this note a new class of topological spaces generalizingk-spaces, the pseudo-k-spaces, is introduced and investigated. Particu-lar attention is given to the study of products of such spaces, in analogyto what is already known aboutk-spaces and quasi-k-spacesMiranda, AM. (2008). On pseudo-k-spaces. Applied General Topology. 9(2):177-184. doi:10.4995/agt.2008.1797.SWORD1771849

Review of Displacements: Women, Tradition, Literatures in French

Reviews the book `Displacements: Women, Tradition, Literatures in French,\u27 edited by Joan DeJean and Nancy K. Miller

Few remarks on maximal pseudocompactness

[EN] A pseudocompact space is maximal pseudocompact if every strictly finer topology is no longer pseudocompact. The main result here is a counterexample which answers a question rised by Alas, Sanchis and Wilson.The author thanks Richard Wilson for the useful discussion concerning thepresentation of Example2.9. A great thank also to the referee for the usefulcomments and the careful reading. This research was partially supported by agrant of the group GNSAGA of INdAM.Bella, A. (2018). Few remarks on maximal pseudocompactness. Applied General Topology. 19(1):155-161. doi:10.4995/agt.2018.7888SWORD15516119