Existence, regularity and boundary behaviour of bounded variation solutions of a one-dimensional capillarity equation

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Classical and non-classical sign changing solutions of a one-dimensional autonomous prescribed curvature equation.

We discuss existence and multiplicity of solutions of the one-dimensional autonomous prescribed curvature problem $$-\left( {u'}/{\sqrt{1+{u'}^2}}\right)' = f(u), \quad u(0)=0,\,\,u(1)=0,$$ depending on the behaviour at the origin and at infinity of the function $f$. We consider solutions that are possibly discontinuous at the points where they attain the value zero

Some notes on weakly Whyburn spaces.

We construct in ZFC two compact weakly Whyburn spaces that are not hereditarily weakly Whyburn. We also construct a Hausdorff countably compact space and a Tychonoff topological group both of weight $\omega_1$ that are not weakly Whyburn. We finally show that Whyburn and weakly Whyburn properties are not preserved by pseudo-open maps

Continuous images of H* and its subcontinua.

We give new sufficient conditions for a continuum to be a remainder of $H$. We also show that any non-degenerate subcontinuum of $H^*$ maps onto any continuum of weight $\leq\omega_1$, thus generalizing a result of D.~P.~Bellamy

When is a topology on a product a product topology?

We study the following problem: when is a given topology \TT on a cartesian product $X\times Y$ a product topology of two topologies \TT_X and \TT_Y on its factors? We are interested in particular in the case when \TT is a weak topology

On compactifications of the set of natural numbers and the half line

The thesis is divided into two distinct parts. In the first part we raise some questions about pseudoradial and related spaces in the setting of the compactifications of N. In the second part we investigate the class of the remainders of H (the half-open interval (0, 1)), we show that some spaces can always be remainders of H and some others (including all the continua of weight \le\omega\sb1) are continuous images of any non-degenerate subcontinuum of $H\sp*.

Open maps do not preserve Whyburn properties.

We show that a (weakly) Whyburn space $X$ may be mapped continuously via an open map $f$ onto a non (weakly) Whyburn space $Y$ . This fact may happen even between topological groups $X$ and $Y$, $f$ a homomorphism, $X$ Whyburn and $Y$ not even weakly Whyburn

Independent-type structures and the number of closed subsets of a space.

Three different notions of an independent family of sets are considered, and it is shown that they are all equivalent under certain conditions. In particular it is proved that in a compact space $X$ in which there is a dyadic system of size $\tau$ there exists also an independent matrix of closed subsets of size $\tau\times 2^\tau$. The cardinal function $M(X,\kappa)$ counting the number of disjoint closed subsets of size larger than or equal to $\kappa$ is introduced and some of its basic properties are studied