Let (X, T ) be a topological dynamical system (TDS), and h(T, K) the topological entropy of a subset K of X . (X, T ) is lowerable if for each 0 ≤ h ≤ h(T, X) there is a non-empty compact subset with entropy h; is hereditarily lowerable if each non-empty compact subset is lowerable; is hereditarily uniformly lowerable if for each non-empty compact subset K and each 0 ≤ h ≤ h(T, K) there is a no...

A space X is submaximal if any dense subset of open. In this paper, we prove that every topological gyrogroup non-measurable cardinality strongly ?-discrete. Moreover, hereditarily paracompact.

In this paper we use a natural forcing to construct a left-separated topology on an arbitrary cardinal κ. The resulting left-separated space Xκ is also 0-dimensional T2, hereditarily Lindelöf, and countably tight. Moreover if κ is regular then d(Xκ) = κ, hence κ is not a caliber of Xκ, while all other uncountable regular cardinals are. This implies that some results of [A] and [JSz] are, consis...

The aim of this note is a classification of all nice and all inductively factored reflection arrangements. It turns out that apart from the supersolvable instances only the monomial groups G(r, r, 3) for r > 3 give rise to nice reflection arrangements. As a consequence of this and of the classification of all inductively free reflection arrangements from Hoge and Röhrle (2015) we deduce that th...

Assuming Jensen’s principle ♦, there is a compact Hausdorff space X which is hereditarily Lindelöf, hereditarily separable, and connected, such that no closed subspace of X is both perfect and totally disconnected. The Proper Forcing Axiom implies that there is no such space. The ♦ example also fails to satisfy the CSWP (the complex version of the Stone-Weierstrass Theorem). This space cannot c...