Tightness of compact spaces is preserved by the tt-equivalence relation

summary:We prove that if there is an open mapping from a subspace of $C_p(X)$ onto $C_p(Y)$, then $Y$ is a countable union of images of closed subspaces of finite powers of $X$ under finite-valued upper semicontinuous mappings. This allows, in particular, to prove that if $X$ and $Y$ are $t$-equivalent compact spaces, then $X$ and $Y$ have the same tightness, and that, assuming $2^{\frak t}>\frak c$, if $X$ and $Y$ are $t$-equivalent compact spaces and $X$ is sequential, then $Y$ is sequential

On some classes of Lindel\"of Sigma-spaces

We consider special subclasses of the class of Lindel\"of Sigma-spaces obtained by imposing restrictions on the weight of the elements of compact covers that admit countable networks: A space $X$ is in the class $L\Sigma(\leq\kappa)$ if it admits a cover by compact subspaces of weight $\kappa$ and a countable network for the cover. We restrict our attention to $\kappa\leq\omega$. In the case $\kappa=\omega$, the class includes the class of metrizably fibered spaces considered by Tkachuk, and the $P$-approximable spaces considered by Tkacenko. The case $\kappa=1$ corresponds to the spaces of countable network weight, but even the case $\kappa=2$ gives rise to a nontrivial class of spaces. The relation of known classes of compact spaces to these classes is considered. It is shown that not every Corson compact of weight $\aleph_1$ is in the class $L\Sigma(\leq \omega)$, answering a question of Tkachuk. As well, we study whether certain compact spaces in $L\Sigma(\leq\omega)$ have dense metrizable subspaces, partially answering a question of Tkacenko. Other interesting results and examples are obtained, and we conclude the paper with a number of open questions.Comment: 21 pages. to appear in Topology and its Application

Beobachtungen und numerische Simulationen polarer Fackeln der Sonne.

In dieser Arbeit werden anhand von hochaufgelösten spektropolaritmetrischen Beobachtungen und numerischen Sumilationen die Struktur und Dynamik von polaren Fackeln der Sonne untersucht, die für das Verständnis des solaren Magnetzyklus wichtig sind.Kilo-Gauss Magnetfelder der dominierenden magnetischen Polarität in polaren Fackeln bestätigen die Annahme, dass diese zum globalen poloidalen Sonnenmagnetfeld gehören. Die komplexe Feinstruktur polarer Fackeln ändert sich innerhalb von 50 Sekunden. Die unerwartete Entdeckung systematischer Aufwärtsströmungen in polaren Fackeln weist auf eine mögliche Verbindung von photosphärischen Fackeln in größeren Breiten mit polaren koronalen Löchern hin, die Quelle des schnellen Sonnenwindes sind.Um eine Hypothese, in der polare Fackeln als Konzentration kleinskaliger magnetischer Flussröhren angenommen werden, zu bestätigen, wurde eine neuartige numerische Simulation durchgeführt. Multi-ray 1.5D Strahlungstransportrechnungen wurden entlang schräger Sehstrahlen durch eine hoch inhomogene Atmosphäre durchgeführt. Die freien Paramter des Modelles wurden so bestimmt dass das Modell den Beobachtungen genügt.In this work the structure and dynamics of polar faculae on the Sun, which constitute an important part of the magnetic solar cycle, were studied by means of high resolution spectropolarimetric observations and numerical simulation.Kilo-Gauss magnetic fields of dominating polarity in polar faculae confirm the concept that they belong to the global poloidal magnetic field of the Sun. Complex fine structure of faculae changes noticeably within 50 s. The unexpected discovery of systematic upflows in polar faculae points at the possible relation of photospheric faculae at high latitudes of the Sun to the polar coronal holes as sources of the fast solar wind from the polar caps.To verify a hypothesis about polar faculae as a concentration of small-scale magnetic flux tubes a numerical simulation was carried out. Multi-ray 1.5D radiative transfer calculations were performed along oblique rays going through a highly inhomogeneous atmosphere of the 3D simulation box. A set of free parameters of the model was deduced, which satisfy observational constraints

On analyticity in cosmic spaces

summary:We prove that a cosmic space (= a Tychonoff space with a countable network) is analytic if it is an image of a $K$-analytic space under a measurable mapping. We also obtain characterizations of analyticity and $\sigma$-compactness in cosmic spaces in terms of metrizable continuous images. As an application, we show that if $X$ is a separable metrizable space and $Y$ is its dense subspace then the space of restricted continuous functions $C_p(X\mid Y)$ is analytic iff it is a $K_{\sigma \delta }$-space iff $X$ is $\sigma$-compact

A remark on the tightness of products

summary:We observe the existence of a $\sigma$-compact, separable topological group $G$ and a countable topological group $H$ such that the tightness of $G$ is countable, but the tightness of $G\times H$ is equal to $\frak c$

Fréchet property in compact spaces is not preserved by MM-equivalence

summary:An example of two $M$-equivalent (hence $l$-equivalent) compact spaces is presented, one of which is Fréchet and the other is not

Tightness of compact spaces is preserved by the tt-equivalence relation

summary:We prove that if there is an open mapping from a subspace of $C_p(X)$ onto $C_p(Y)$, then $Y$ is a countable union of images of closed subspaces of finite powers of $X$ under finite-valued upper semicontinuous mappings. This allows, in particular, to prove that if $X$ and $Y$ are $t$-equivalent compact spaces, then $X$ and $Y$ have the same tightness, and that, assuming $2^{\frak t}>\frak c$, if $X$ and $Y$ are $t$-equivalent compact spaces and $X$ is sequential, then $Y$ is sequential