Erdős space $\mathfrak {E}$ and complete Erdős space $\mathfrak {E}_{c}$ have been previously shown to have topological characterizations. In this paper, we provide a topological characterization of the topological space $\mathbb {Q}\times \mathfrak {E}_{c}$ , where $\mathbb {Q}$ is the space of rational numbers. As a corollary, we show that the Vietoris hyperspace of finite sets $\mathcal {F}(\mathfrak {E}_{c})$ is homeomorphic to $\mathbb {Q}\times \mathfrak {E}_{c}$ . We also characterize the factors of $\mathbb {Q}\times \mathfrak {E}_{c}$ . An interesting open question that is left open is whether $\sigma \mathfrak {E}_{c}^{\omega }$ , the $\sigma $ -product of countably many copies of $\mathfrak {E}_{c}$ , is homeomorphic to $\mathbb {Q}\times \mathfrak {E}_{c}$ .

For a continuous function $f:[0,1] \to [0,1]$ we define a splitting sequence admitted by f and show that the inverse limit of f is an arc if and only if f does not admit a splitting sequence.

For each discriminant $D>1$, McMullen constructed the Prym–Teichmüller curves $W_{D}(4)$ and $W_{D}(6)$ in ${\mathcal{M}}_{3}$ and ${\mathcal{M}}_{4}$, which constitute one of the few known infinite families of geometrically primitive Teichmüller curves. In the present paper, we determine for each $D$ the number and type of orbifold points on $W_{D}(6)$. These results, together with a previous result of the two authors in the genus $3$ case and with results of Lanneau–Nguyen and Möller, complete the topological characterisation of all Prym–Teichmüller curves and determine their genus. The study of orbifold points relies on the analysis of intersections of $W_{D}(6)$ with certain families of genus $4$ curves with extra automorphisms. As a side product of this study, we give an explicit construction of such families and describe their Prym–Torelli images, which turn out to be isomorphic to certain products of elliptic curves. We also give a geometric description of the flat surfaces associated to these families and describe the asymptotics of the genus of $W_{D}(6)$ for large $D$.

Continua $X$ and $Y$ are monotone equivalent if there exist monotone onto maps $f\,:\,X\,\to \,Y$ and $g:\,Y\to \,X.\,\text{A}$ . A continuum $X$ is isolated with respect to monotone maps if every continuumthat is monotone equivalent to $X$ must also be homeomorphic to $X$ . In this paper we show that a dendrite $X$ is isolated with respect to monotone maps if and only if the set of ramification points of $X$ is finite. In this way we fully characterize the classes of dendrites that are monotone isolated.

In 1940 Paul Erdős introduced the ‘rational Hilbert space’, which consists of all vectors in the real Hilbert space $\ell^2$ that have only rational coordinates. He showed that this space has topological dimension one, yet it is totally disconnected and homeomorphic to its square. In this note we generalize the construction of this peculiar space and we consider all subspaces $\mathcal{E}$ of the Banach spaces $\ell^p$ that are constructed as ‘products’ of zero-dimensional subsets $E_n$ of $\mathbb{R}$. We present an easily applied criterion for deciding whether a general space of this type is one dimensional. As an application we find that if such an $\mathcal{E}$ is closed in $\ell^p$, then it is homeomorphic to complete Erdős space if and only if $\dim\mathcal{E}>0$ and every $E_n$ is zero dimensional.

It is proved, in particular, that a topological space X is a Baire space if and only if every real valued function f: X →R is almost continuous on a dense subset of X. In fact, in the above characterization of a Baire space, the range space R of real numbers may be generalized to any second countable, Hausdorfï space that contains infinitely many points.