We construct a completely regular space which is connected, locally connected and countable dense homogeneous but not strongly locally homogeneous. The space has an open subset which has a unique cut-point. We use the construction of a ${C}^{1}$-diffeomorphism of the plane which takes one countable dense set to another.

For non-empty topological spaces X and Y and arbitrary families $\mathcal{A}$ ⊆ $\mathcal{P}\left(X\right)$ and $\mathcal{B}\subseteq \mathcal{P}\left(Y\right)$ we put ${\mathcal{C}}_{\mathcal{A},\mathcal{B}}$=f ∈ ${Y}^{X}$ : (∀ A ∈ $\mathcal{A}$)(f[A] ∈ $\mathcal{B})$. We examine which classes of functions $\mathcal{F}$ ⊆ ${Y}^{X}$ can be represented as ${\mathcal{C}}_{\mathcal{A},\mathcal{B}}$. We are mainly interested in the case when $\mathcal{F}=\mathcal{C}(X,Y)$ is the class of all continuous functions from X into Y. We prove that for a non-discrete Tikhonov space X the class $\mathcal{F}=\mathcal{C}$(X,ℝ) is not equal to ${\mathcal{C}}_{\mathcal{A},\mathcal{B}}$ for any $\mathcal{A}$ ⊆ $\mathcal{P}\left(X\right)$ and $\mathcal{B}$ ⊆ $\mathcal{P}$(ℝ). Thus, $\mathcal{C}$(X,ℝ) cannot be characterized by images of sets. We also show that none of the following classes of...

Costruiamo uno spazio nontransitivo analogo al piano di Kofner. Mentre gli argomenti usati per la costruzione del piano di Kofner si fondano su riflessioni geometriche, le nostre prove si basano su idee combinatorie.

For a cardinal $\alpha $, we say that a subset $B$ of a space $X$ is ${C}_{\alpha}$-compact in $X$ if for every continuous function $f\phantom{\rule{0.222222em}{0ex}}X\to {\mathbb{R}}^{\alpha}$, $f\left[B\right]$ is a compact subset of ${\mathbb{R}}^{\alpha}$. If $B$ is a $C$-compact subset of a space $X$, then $\rho (B,X)$ denotes the degree of ${C}_{\alpha}$-compactness of $B$ in $X$. A space $X$ is called $\alpha $-pseudocompact if $X$ is ${C}_{\alpha}$-compact into itself. For each cardinal $\alpha $, we give an example of an $\alpha $-pseudocompact space $X$ such that $X\times X$ is not pseudocompact: this answers a question posed by T. Retta in “Some cardinal generalizations of pseudocompactness”...

We construct a completely regular ordered space $(X,\mathcal{T},\le )$ such that $X$ is an $I$-space, the topology $\mathcal{T}$ of $X$ is metrizable and the bitopological space $(X,{\mathcal{T}}^{\u266f},{\mathcal{T}}^{\u266d})$ is pairwise regular, but not pairwise completely regular. (Here ${\mathcal{T}}^{\u266f}$ denotes the upper topology and ${\mathcal{T}}^{\u266d}$ the lower topology of $X$.)

It is proved that every non trivial continuous map between the sets of extremal elements of monotone sequential cascades can be continuously extended to some subcascades. This implies a result of Franklin and Rajagopalan that an Arens space cannot be continuously non trivially mapped to an Arens space of higher rank. As an application, it is proved that if for a filter $\mathscr{H}$ on $\omega $, the class of $\mathscr{H}$-radial topologies contains each sequential topology, then it includes the class of subsequential topologies....

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