We prove that if a space X is countable dense homogeneous and no set of size n-1 separates it, then X is strongly n-homogeneous. Our main result is the construction of an example of a Polish space X that is strongly n-homogeneous for every n, but not countable dense homogeneous.
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We present an example of a connected, Polish, countable dense homogeneous space X that is not strongly locally homogeneous. In fact, a nontrivial homeomorphism of X is the identity on no nonempty open subset of X.
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We show that there is a Polish space which is countable dense homogeneous but contains a dense open rigid connected subset. This answers several questions of Fitzpatrick and Zhou.
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If X is a space then L(X) denotes the subspace of C(X) consisting of all Peano (sub)continua. We prove that for n ≥ 3 the space $L(ℝ^n)$ is homeomorphic to $B^∞$, where B denotes the pseudo-boundary of the Hilbert cube Q.
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We prove that $trt K_ω > ω+1$, where trt stands for the transfinite extension of Steinke's separation dimension. This answers a question of Chatyrko and Hattori.
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We prove that the statement: "there is a Corson compact space with a non-separable Radon measure" is equivalent to a number of natural statements in set theory.
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We investigate the following question: under which conditions is a σ-compact partial two point set contained in a two point set? We show that no reasonable measure or capacity (when applied to the set itself) can provide a sufficient condition for a compact partial two point set to be extendable to a two point set. On the other hand, we prove that under Martin's Axiom any σ-compact partial two point set such that its square has Hausdorff 1-measure zero is extendable.
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We show that the assertions of some fundamental selection theorems for lower-semicontinuous maps with completely metrizable range and metrizable domain actually characterize topological completeness of the target space. We also show that certain natural restrictions on the class of the domains change this situation. The results provide in particular answers to questions asked by Engelking, Heath and Michael [3] and Gutev, Nedev, Pelant and Valov [5].
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We show that all sufficiently nice λ-sets are countable dense homogeneous (𝖢𝖣𝖧). From this fact we conclude that for every uncountable cardinal κ ≤ 𝔟 there is a countable dense homogeneous metric space of size κ. Moreover, the existence of a meager in itself countable dense homogeneous metric space of size κ is equivalent to the existence of a λ-set of size κ. On the other hand, it is consistent with the continuum arbitrarily large that every 𝖢𝖣𝖧 metric space has size either ω₁ or 𝔠. An example of a Baire 𝖢𝖣𝖧 metric space which is not completely metrizable is presented. Finally, answering a question of Arhangel'skii and van Mill we show that that there is a compact non-metrizable 𝖢𝖣𝖧 space in ZFC.
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