On almost discrete space

summary:Let $C(X)$ be the ring of real continuous functions on a completely regular Hausdorff space. In this paper an almost discrete space is determined by the algebraic structure of $C(X)$. The intersection of essential weak ideal in $C(X)$ is also studied

Cozero Complemented Spaces; When the Space of Minimal Prime Ideals of a C(X) is Compact

If X is a Tychonoff space, C(X) its ring of real-valued continuous functions, and f C(X), then the cozeroset of f is coz(f)= {xX: f(x)≠0}. If, for every cozeroset V of X, there is a disjoint cozeroset V′ such that V V′ is dense in X, then X is said to be cozero complemented. It has long been known that X is cozero complemented iff the space MinC(X) of minimal prime ideals of C(X) (in the hull-kernel or Zariski topology) is compact iff the classical ring of fractions of C(X) is von Neumann regular. While many characterizations of cozero complemented spaces are known, they seem not to be adequate to answer some natural questions about them raised by R. Levy and J. Shapiro in an unpublished preprint. These questions concern the relationship between a space being cozero complemented and certain kinds of subspaces having this property, and between a product of two spaces being cozero complemented and the factor spaces being cozero complemented. Also, some conditions are given that guarantee that a space that is locally cozero complemented has this property globally. In this paper partial answers are given to these questions. Sample results: If X is weakly Lindelöf and dense in T, then X is cozero complemented iff T is cozero complemented; if X×Y is weakly Lindelöf and cozero complemented, then X and Y are cozero complemented, but if D is an uncountable discrete space, then βD×βD is not cozero complemented even though βD is cozero complemented. If X is locally cozero complemented and either weakly Lindelöf or paracompact, then X is cozero complemented

From peculiar morphologies to Hubble-type spirals: the relation between galaxy dynamics and morphology in star-forming galaxies at z ∼ 1.5

We present an analysis of the gas dynamics of star–forming galaxies at z ∼ 1.5 using data from the KMOS Galaxy Evolution Survey (KGES). We quantify the morphology of the galaxies using HSTCANDELS imaging parametrically and non-parametrically. We combine the Hα dynamics from KMOS with the high–resolution imaging to derive the relation between stellar mass (M*) and stellar specific angular momentum (j*). We show that high–redshift star–forming galaxies at z ∼ 1.5 follow a power-law trend in specific stellar angular momentum with stellar mass similar to that of local late–type galaxies of the form j* ∝ M0.53±0.10∗⁠. The highest specific angular momentum galaxies are mostly disc–like, although generally, both peculiar morphologies and disc-like systems are found across the sequence of specific angular momentum at a fixed stellar mass. We explore the scatter within the j* – M* plane and its correlation with both the integrated dynamical properties of a galaxy (e.g. velocity dispersion, Toomre Qg, Hα star formation rate surface density ΣSFR) and its parameterised rest-frame UV / optical morphology (e.g. Sérsic index, bulge to total ratio, Clumpiness, Asymmetry and Concentration). We establish that the position in the j* – M* plane is strongly correlated with the star-formation surface density and the Clumpiness of the stellar light distribution. Galaxies with peculiar rest-frame UV / optical morphologies have comparable specific angular momentum to disc – dominated galaxies of the same stellar mass, but are clumpier and have higher star-formation rate surface densities. We propose that the peculiar morphologies in high–redshift systems are driven by higher star formation rate surface densities and higher gas fractions leading to a more clumpy inter-stellar medium

The Shapes of the Rotation Curves of Star-forming Galaxies Over the Last ≈ 10 Gyr

We analyse maps of the spatially-resolved nebular emission of ≈1500 star-forming galaxies at z ≈ 0.6–2.2 from deep KMOS and MUSE observations to measure the average shape of their rotation curves. We use these to test claims for declining rotation curves at large radii in galaxies at z ≈ 1–2 that have been interpreted as evidence for an absence of dark matter. We show that the shape of the average rotation curves, and the extent to which they decline beyond their peak velocities, depends upon the normalisation prescription used to construct the average curve. Normalising in size by the galaxy stellar disk-scale length after accounting for seeing effects (⁠R′d⁠), we construct stacked position-velocity diagrams that trace the average galaxy rotation curve out to 6R′d (≈13 kpc, on average). Combining these curves with average HI rotation curves for local systems, we investigate how the shapes of galaxy rotation curves evolve over ≈10 Gyr. The average rotation curve for galaxies binned in stellar mass, stellar surface mass density and/or redshift is approximately flat, or continues to rise, out to at least 6R′d⁠. We find a trend between the outer slopes of galaxies’ rotation curves and their stellar mass surface densities, with the higher surface density systems exhibiting flatter rotation curves. Drawing comparisons with hydrodynamical simulations, we show that the average shapes of the rotation curves for our sample of massive, star-forming galaxies at z ≈ 0–2.2 are consistent with those expected from ΛCDM theory and imply dark matter fractions within 6Rd of at least ≈60 percent