General Topology
The paper investigates the variation of the spectrum of operators in infinite dimensional Banach spaces. In particular, it is shown that the spectrum function is Borel from the space of bounded operators on a separable Banach space;…
Techniques from Descriptive Set Theory are applied in order to study the Topological Complexity of families of operators naturally connected to ergodic operators in infinite dimensional Banach Spaces. The families of ergodic,…
In addition to 29 announcements in related areas, this issue contains several contributions to "core" SPM: Measurable cardinals and the cardinality of Lindelof spaces; Topological games and covering dimension; Menger's and Hurewicz's…
The natural quotient map q from the space of based loops in the Hawaiian earring onto the fundamental group provides a new example of a quotient map such that q x q fails to be a quotient map. This also settles in the negative the question…
In a recent paper by D. Shakhmatov and J. Sp\v{e}v\'ak [Group-valued continuous functions with the topology of pointwise convergence, Topology and its Applications (2009), doi:10.1016/j.topol.2009.06.022] the concept of a ${\rm TAP}$ group…
In this brief note we provide a simple approach to give a new proof of the well known fact that the Banach-Alaoglu theorem and the Tychonoff product theorem for compact Hausdorff spaces are equivalent.
A Hausdorff topological group G is minimal if every continuous isomorphism f: G --> H between G and a Hausdorff topological group H is open. Significantly strengthening a 1981 result of Stoyanov, we prove the following theorem: For every…
For an abelian topological group G let G^* denote the dual group of all continuous characters endowed with the compact open topology. Given a closed subset X of an infinite compact abelian group G such that w(X) < w(G) and an open…
This paper has been withdrawn by the author. Much simpler proof of the main result was obtained which led to major changes in the presentation.
Assuming the absence of Q-points (which is consistent with ZFC) we prove that the free topological group $F(X)$ over a Tychonov space $X$ is $o$-bounded if and only if every continuous metrizable image $T$ of $X$ satisfies the selection…
We investigate certain geometric properties of the spaces of idempotent measures. In particular, we prove that the space of idempotent measures on an infinite compact metric space is homeomorphic to the Hilbert cube.
Our main result states that the hyperspace of convex compact subsets of a compact convex subset $X$ in a locally convex space is an absolute retract if and only if $X$ is an absolute retract of weight $\le\omega_1$. It is also proved that…
We give a construction under $CH$ of a non-metrizable compact Hausdorff space $K$ such that any uncountable semi-biorthogonal sequence in $C(K)$ must be of a very specific kind. The space $K$ has many nice properties, such as being…
Necessary and sufficient conditions for a finite connected graph with a strict partial order on vertices to be a combinatorial invariant of pseudoharmonic function are obtained.
Let $D^2 \subset C$ be a closed two-dimensional disk and $f:D^2 \to R$ be a continuous function such that a restriction of $f$ to $\partial D^2$ is a continuous function with a finite number of local extrema and $f$ has a finite number of…
Generalizing de Vries Compactification Theorem and strengthening Leader Local Compactification Theorem, we describe the partially ordered set $(\LL(X),\le)$ of all (up to equivalence) locally compact Hausdorff extensions of a Tychonoff…
Generalizing a theorem of Ph. Dwinger, we describe the partially ordered set of all (up to equivalence) zero-dimensional locally compact Hausdorff extensions of a zero-dimensional Hausdorff space. Using this description, we find the…
We introduce a class of regular continuous functions on the closed 2-disk and show that each function from this class is topologically conjugate to a linear function defined on a sqare, a closed half-disk or a closed disk.
We investigate contrasting behaviours emerging when studying foliations on non-metrisable manifolds. It is shown that Kneser's pathology of a manifold foliated by a single leaf cannot occur with foliations of dimension-one. On the other…
We investigate the mapping class group of an orientable $\omega$-bounded surface. Such a surface splits, by Nyikos's Bagpipe Theorem, into a union of a bag (a compact surface with boundary) and finitely many long pipes. The subgroup…