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We use the H$41\alpha$ recombination line to create templates of the millimeter free-free emission in the ALMA-IMF continuum maps, which allows to separate it from dust emission. This method complements spectral-index information and extrapolation from centimeter wavelength maps. We use the derived maps to estimate the properties of up to 34 HII regions across the ALMA-IMF protoclusters. The hydrogen ionizing-photon rate $Q_0$ and spectral types follow the evolutionary trend proposed by Motte et al. The youngest protoclusters lack detectable ionized gas, followed by protoclusters with increasing numbers of OB stars. The total $Q_0$ increases from $\sim 10^{45}$ s$^{-1}$ to $> 10^{49}$ s$^{-1}$. We used the adjacent He$41\alpha$ line to measure the relative number abundances of helium, finding values consistent with the Galactic interstellar medium, although a few outliers are discussed. A search for sites of maser amplification of the H$41\alpha$ line returned negative results. We looked for possible correlations between the electron densities ($n_e$), emission measures (EM), and $Q_0$ with HII region size $D$. The latter are the better correlated, with $Q_0 \propto D^{2.49\pm0.18}$. This favors interpretations where smaller ultracompact HII regions are not necessarily the less dynamically evolved versions of larger ones, but rather are ionized by less massive stars. Moderate correlations were found between dynamical width $\Delta V_\mathrm{dyn}$ with $D$ and $Q_0$. $\Delta V_\mathrm{dyn}$ increases from about one to two times the ionized-gas sound speed. Finally, an outlier HII region south of W43-MM2 is discussed. We suggest that this source could harbor an embedded stellar or disk wind., Comment: Accepted to AAS Journals (ApJS). Data available at https://dataverse.harvard.edu/dataverse/h41a-freefree

The cellular-Lindel\"of property is a common generalization of the Lindel\"of property and the countable chain condition that was introduced by Bella and Spadaro in 2018. We solve two questions of Alas, Gutierrez-Dominguez and Wilson by constructing consistent examples of a normal almost cellular-Lindel\"of space which is neither cellular-Lindel\"of nor weakly Lindel\"of and a Tychonoff cellular-Lindel\"of space of Lindel\"of degree $\omega_1$ and uncountable weak Lindel\"of degree for closed sets. We also construct a ZFC example of a space for which both the cellular-Lindel\"of property and normality are undetermined in ZFC.

A space is functionally countable if every real-valued continuous function has countable image. A stronger property recently defined by Tkachuk is exponentially separability. We start by studying these properties in GO spaces, where we extend results by Tkachuk and Wilson, and prove a conjecture by Dow. We also study some subspaces of products that are functionally countable. Finally, we give some examples of functionally countable spaces that are separable and uncountable.

We present the combination of ALMA-IMF and single-dish continuum images from the Mustang-2 Galactic Plane Survey (MGPS90) at 3 millimeters and the Bolocam Galactic Plane Survey (BGPS) at 1 millimeter. Six and ten out of the fiffteen ALMA-IMF fields are combined with MGPS90 and BGPS, respectively. The combination is made via the feathering technique. We used the dendrogram algorithm throughout the combined images, and performed further analysis in the six fields with combination in both bands (G012.80, W43-MM1, W43-MM2, W43-MM3, W51-E, W51-IRS2). In these fields, we calculated spectral index maps and used them to separate regions dominated by dust or free-free emission, and then performed further structural analysis. We report the basic physical parameters of the dust-dominated (column densities, masses) and ionized (emission measures, hydrogen ionization photon rates) structures. We also searched for multi-scale relations in the dust-dominated structures across the analyzed fields, finding that the fraction of mass in dendrogram leaves (which we label as "Leaf Mass Eficiency", LME) as a function of molecular gas column density follows a similar trend: a rapid, exponential-like growth, with maximum values approaching 100% in most cases. The observed behaviour of the LME with gas column is tentatively interpreted as an indicator of large star formation activity within the ALMA-IMF protoclusters. W51-E and G012.80 stand out as cases with comparatively large and reduced potential for further star formation, respectively., Comment: Accepted to The Astrophysical Journal Supplement

In a 2021 paper, Vladimir Tkachuk asked whether there is a non-separable LOTS $X$ such that $X^2\setminus\{\langle x,x\rangle\colon x\in X\}$ is functionally countable. In this paper we prove that such a space, if it exists, must be an Aronszajn line and admits a $\leq 2$-to-$1$ retraction to a subspace that is a Suslin line. After this, assuming the existence of a Suslin line, we prove that there is Suslin line that is functionally countable. Finally, we present an example of a functionally countable Suslin line $L$ such that $L^2\setminus\{\langle x,x\rangle\colon x\in L\}$ is not functionally countable.

Let $X$ be a smooth fan and denote its set of endpoints by $E(X)$. Let $E$ be one of the following spaces: the natural numbers, the irrational numbers, or the product of the Cantor set with the natural numbers. We prove that there is a smooth fan $X$ such that $E(X)$ is homeomorphic to $E$ and for every homeomorphism $h \colon X \to X$, the restriction of $h$ to $E(X)$ is the identity. On the other hand, we also prove that if $X$ is any smooth fan such that $E(X)$ is homeomorphic to complete Erd\H{o}s space, then $X$ is necessarily homeomorphic to the Lelek fan; this adds to a 1989 result by W{\l}odzimierz Charatonik., Comment: 15 pages, 4 figures

Given a continuum $X$, let $C(X)$ denote the hyperspace of all subcontinua of $X$. In this paper we study the Vietoris hyperspace $NC^{*}(X)=\{ A \in C(X):X\setminus A\text{ is connected}\}$ when $X$ is a finite graph or a dendrite; in particular, we give conditions under which $NC^{*}(X)$ is compact, connected, locally connected or totally disconnected. Also, we prove that if $X$ is a dendrite and the set of endpoints of $X$ is dense, then $NC^{*}(X)$ is homeomorphic to the Baire space of irrational numbers.

We have identified 453 compact dense cores in 3 mm continuum emission maps in the ATOMS (ALMA Three-millimeter Observations of Massive Star-forming regions) survey, and compiled three catalogues of high-mass star forming cores. One catalogue, referred to as H/UC-HII catalogue, includes 89 cores that enshroud hyper/ultra compact (H/UC) HII regions as characterized by associated compact H40alpha emission. A second catalogue, referred to as pure s-cHMC, includes 32 candidate Hot Molecular Cores (HMCs) showing rich spectra (N>20lines) of complex organic molecules (COMs) but not associated with H/UC-HII regions. The third catalogue, referred to as pure w-cHMC, includes 58 candidate HMCs with relatively low levels of COM richness and not associated with H/UC-HII regions. These three catalogues of dense cores provide an important foundation for future studies of the early stages of high-mass star formation across the Milky Way. We also find that nearly half of H/UC-HII cores are candidate HMCs. From the number counts of COM-containing and H/UC-HII cores, we suggest that the duration of high-mass protostellar cores showing chemically rich features is at least comparable to the lifetime of H/UC-HII regions. For cores in the H/UC-HII catalogue, the width of the H40alpha line increases as the core size decreases, suggesting that the non-thermal dynamical and/or pressure line-broadening mechanisms dominate on the smaller scales of the H/UC-HII cores., Comment: 17 pages, five tables, and 11 figures. Accepted for publication at MNRAS

Erd\H{o}s space $\mathfrak{E}$ and complete Erd\H{o}s 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 $\mathbb{Q}\times\mathfrak{E}_c$.

We analyze the gas mass distribution, the gas kinematics, and the young stellar object (YSO) content of the California Molecular Cloud (CMC) L1482 filament. We derive a Gaia DR2 YSO distance of 511$^{+17}_{-16}$ pc. We derive scale-free power-laws for the mean gas line-mass (M/L) profiles; we calculate the gravitational potential and field profiles consistent with these. We present IRAM 30 m C$^{18}$O (1-0) (and other tracers) position-velocity (PV) diagrams that exhibit complex velocity twisting and turning structures. We find a rotational profile in C$^{18}$O perpendicular to the southern filament ridgeline. The profile is regular, confined ($r\lesssim0.4$ pc), anti-symmetric, and to first order linear with a break at $r\sim0.25$ pc. The timescales of the inner (outer) gradients are $\sim$0.7 (6.0) Myr. We show that the centripetal force, compared to gravity, increases toward the break; when the ratio of forces approaches unity, the profile turns over, just before filament breakup is achieved. The timescales and relative roles of gravity to rotation indicate that the structure is stable, long lived ($\sim$ a few times 6 Myr), and undergoing outside-in evolution. Moreover, this filament has practically no star formation, a perpendicular Planck plane-of-the-sky (POS) magnetic field morphology, and POS "zig-zag" morphology, which together with the rotation profile lead to the suggestion that the 3D shape is a corkscrew filament with a helical magnetic field. These results, combined with results in Orion and G035.39-00.33, suggest evolution toward higher densities as rotating filaments shed angular momentum. Thus, magnetic fields may be an essential feature of high-mass (M $\sim10^5$ M$_{\odot}$) cloud filament evolution toward cluster formation., Comment: 23 pages, 17 figures. Accepted for publication in the Astrophysical Journal

In this paper we consider the question of when the space $C_p(X)$ of continuous real-valued functions on $X$ with the pointwise convergence topology is countable dense homogeneous. In particular, we focus on the case when $X$ is countable with a unique non-isolated point $\infty$. In this case, $C_p(X)$ is countable dense homogeneous if and only if the filter of open neighborhoods of $\infty$ is a non-meager $P$-filter.

In this paper we compare the concepts of pseudoradial spaces and the recently defined strongly pseudoradial spaces in the realm of compact spaces. We show that $\mathrm{MA}+\mathfrak{c}=\omega_2$ implies that there is a compact pseudoradial space that is not strongly pseudoradial. We essentially construct a compact, sequentially compact space $X$ and a continuous function $f:X\to\omega_1+1$ in such a way that there is no copy of $\omega_1+1$ in $X$ that maps cofinally under $f$. We also give some conditions that imply the existence of copies of $\omega_1$ in spaces. In particular, $\mathrm{PFA}$ implies that compact almost radial spaces of radial character $\omega_1$ contain many copies of $\omega_1$.

We show that all sufficiently nice $\lambda$-sets are countable dense homogeneous ($\mathsf{CDH}$). From this fact we conclude that for every uncountable cardinal $\kappa \le \mathfrak{b}$ there is a countable dense homogeneous metric space of size $\kappa$. Moreover, the existence of a meager in itself countable dense homogeneous metric space of size $\kappa$ is equivalent to the existence of a $\lambda$-set of size $\kappa$. On the other hand, it is consistent with the continuum arbitrarily large that every $\mathsf{CDH}$ metric space has size either $\omega_1$ or size $\mathfrak c$. An example of a Baire $\mathsf{CDH}$ 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 $\mathsf{CDH}$ space in ZFC.

Let $\mathbb{A}$ denote the Alexandroff-Urysohn double arrow space. We prove the following results: (a) $\mathbb{A}\times{}^\omega{2}$ is not countable dense homogeneous; (b) ${}^{\omega}{\mathbb{A}}$ is not countable dense homogeneous; (c) $\mathbb{A}$ has exactly $\mathfrak{c}$ types of countable dense subsets. These results answer questions by Arhangel'ski\u\i, Hru\v{s}\'ak and van Mill.

For a given continuum $X$ and a natural number $n,$ we consider the hyperspace $F_n(X)$ of all nonempty subsets of $X$ with at most $n$ points, metrized by the Hausdorff metric. In this paper we show that if $X$ is a dendrite whose set of end points is closed, $n \in \mathbb{N}$ and $Y$ is a continuum such that the hyperspaces $F_n(X)$ and $F_n(Y)$ are homeomorphic, then $Y$ is a dendrite whose set of end points is closed.

We prove that if $\mathcal{F}$ is a non-meager $P$-filter, then both $\mathcal{F}$ and ${}^\omega\mathcal{F}$ are countable dense homogeneous spaces.

Given a Tychonoff space $X$, let $\varrho(X)$ be the set of remote points of $X$. We view $\varrho(X)$ as a topological space. In this paper we assume that $X$ is metrizable and ask for conditions on $Y$ so that $\varrho(X)$ is homeomorphic to $\varrho(Y)$. This question has been studied before by R. G. Woods and C. Gates. We give some results of the following type: if $X$ has topological property $\mathbf{P}$ and $\varrho(X)$ is homeomorphic to $\varrho(Y)$, then $Y$ also has $\mathbf{P}$. We also characterize the remote points of the rationals and irrationals up to some restrictions. Further, we show that $\varrho(X)$ and $\varrho(Y)$ have open dense homeomorphic subspaces if $X$ and $Y$ are both nowhere locally compact, completely metrizable and share the same cellular type, a cardinal invariant we define.

Let $X$ be a Hausdorff space and let $\mathcal{H}$ be one of the hyperspaces $CL(X)$, $\mathcal{K}(X)$, $\mathcal{F}(X)$ or $\mathcal{F}_n(X)$ ($n$ a positive integer) with the Vietoris topology. We study the following disconnectedness properties for $\mathcal{H}$: extremal disconnectedness, being a $F^\prime$-space, $P$-space or weak $P$-space and hereditary disconnectedness. Our main result states: if $X$ is Hausdorff and $F\subset X$ is a closed subset such that $(a)$ both $F$ and $X-F$ are totally disconnected, $(b)$ the quotient $X/F$ is hereditarily disconnected, then $\mathcal{K}(X)$ is hereditarily disconnected. We also show an example proving that this result cannot be reversed.

In a paper by Bella, Tokg\"os and Zdomskyy it is asked whether there exists a Tychonoff space $X$ such that the remainder of $C_p(X)$ in some compactification is Menger but not $\sigma$-compact. In this paper we prove that it is consistent that such space exists and in particular its existence follows from the existence of a Menger ultrafilter.

It is shown that CH implies the existence of a compact Hausdorff space that is countable dense homogeneous, crowded and does not contain topological copies of the Cantor set. This contrasts with a previous result by the author which says that for any crowded Hausdorff space $X$ of countable $\pi$-weight, if ${}^\omega{X}$ is countable dense homogeneous, then $X$ must contain a topological copy of the Cantor set.