Separable Hausdorff Space Research Articles

Comparison radius and mean topological dimension: Rokhlin property, comparison of open sets, and subhomogeneous C*-algebras

Let (X, Γ) be a free minimal dynamical system, where X is a compact separable Hausdorff space and Γ is a discrete amenable group. It is shown that, if (X, Γ) has a version of Rokhlin property (uniform Rokhlin property) and if C(X) ⋊ Γ has a Cuntz comparison on open sets, then the comparison radius of the crossed product C*-algebra C(X) ⋊ Γ is at most half of the mean topological dimension of (X, Γ).These two conditions are shown to be satisfied if Γ = ℤ or if (X, Γ) is an extension of a free Cantor system and Γ has subexponential growth. The main tools being used are Cuntz comparison of diagonal elements of a subhomogeneous C*-algebra and small subgroupoids.

Cardinal invariants of dually CCC spaces

We say that a space X is dually CCC (respectively, weakly Lindelöf, separable) if for any neighbourhood assignment ϕ on X, there is a CCC (respectively, weakly Lindelöf, separable) subspace Y⊂X such that ϕ(Y)={ϕ(y):y∈Y} covers X. In this paper, we mainly show that(1) A dually CCC first countable Hausdorff space has cardinality at most 2c and a dually weakly Lindelöf first countable normal space has cardinality at most 2c.(2) Let Y=∏{Yi:i≤n}, where Yi is a scattered monotonically normal space for any i=0,1,...,n. If a subspace X⊂Y is dually CCC then e(X)≤ω and a normal subspace X⊂Y is DCCC if and only if e(X)≤ω.(3) Assume 2<c=c. A normal dually CCC space X with χ(X)≤c has extent at most c.(4) A dually separable Hausdorff space X with a Gδ⁎-diagonal has extent at most c and a dually separable regular space X with a Gδ-diagonal has cardinality at most c.(5) A dually CCC Hausdorff space with a Gδ-diagonal has cellularity at most c.

A study of chain conditions and dually properties

In this paper, we make some observations on chain conditions and dually properties. In particular, we show that:(1) A subspace X⊂ω1ω is dually CCC then e(X)≤ω and a normal subspace X⊂ω1ω is DCCC if and only if e(X)≤ω;(2) There is a Tychonoff pseudocompact subspace X⊂(ω1+1)2 which is not dually CCC;(3) In the class of o-semimetrizable spaces, dually separable is self-dual with respect to neighbourhood assignments. As an application, we obtain an example of a CCC normal Moore space which is not dually separable under MA+¬CH;(4) There exists an example of a large normal CCC semi-stratifiable space, which answers a question of Xuan and Song (2018) [21, Question 4.11];(5) Every dually separable and monotonically monolithic space is Lindelöf, which gives a partial answer to a question of Alas, Junqueira, van Mill, Tkachuk and Wilson (2011) [2, Question 2.1];(6) A dually separable Hausdorff space with a strong rank 1-diagonal has cardinality at most 2c. The conclusion is also true for regular spaces if we replace “strong rank 1-diagonal” with “Gδ-diagonal”;(7) A dually separable ω-monolithic Hausdorff space with a Gδ-diagonal has cardinality at most c.

The weight and Lindelöf property in spaces and topological groups

We show that if Y is a dense subspace of a Tychonoff space X, then w(X)≤nw(Y)Nag(Y), where Nag(Y) is the Nagami number of Y. In particular, if Y is a Lindelöf Σ-space, then w(X)≤nw(Y)ω≤nw(X)ω.Better upper bounds for the weight of topological groups are given. For example, if a topological group H contains a dense subgroup G such that G is a Lindelöf Σ-space, then w(H)=w(G)≤ψ(G)ω. Further, if a Lindelöf Σ-space X generates a dense subgroup of a topological group H, then w(H)≤2ψ(X).Several facts about subspaces of Hausdorff separable spaces are established. It is well known that the weight of a separable Hausdorff space X can be as big as 22c. We prove on the one hand that if a regular Lindelöf Σ-space Y is a subspace of a separable Hausdorff space, then w(Y)≤2ω, and the same conclusion holds for a Lindelöf P-space Y. On the other hand, we present an example of a countably compact topological Abelian group G which is homeomorphic to a subspace of a separable Hausdorff space and satisfies w(G)=22c, i.e. G has the maximal possible weight.

Density character of subgroups of topological groups

We give a complete characterization of subgroups of separable topological groups. Then we show that the following conditions are equivalent for an ω \omega -narrow topological group G G : (i) G G is homeomorphic to a subspace of a separable regular space; (ii) G G is topologically isomorphic to a subgroup of a separable topological group; (iii) G G is topologically isomorphic to a closed subgroup of a separable path-connected, locally path-connected topological group. A pro-Lie group is a projective limit of finite-dimensional Lie groups. We prove here that an almost connected pro-Lie group is separable if and only if its weight is not greater than the cardinality c \mathfrak {c} of the continuum. It is deduced from this that an almost connected pro-Lie group is separable if and only if it is homeomorphic to a subspace of a separable Hausdorff space. It is also proved that a locally compact (even feathered) topological group G G which is a subgroup of a separable Hausdorff topological group is separable, but the conclusion is false if it is assumed only that G G is homeomorphic to a subspace of a separable Tychonoff space. We show that every precompact (abelian) topological group of weight less than or equal to c \mathfrak {c} is topologically isomorphic to a closed subgroup of a separable pseudocompact (abelian) group of weight c \mathfrak {c} . This result implies that there is a wealth of closed non-separable subgroups of separable pseudocompact groups. An example is also presented under the Continuum Hypothesis of a separable countably compact abelian group which contains a non-separable closed subgroup.

An extension of Kolmogorov’s theorem for continuous covariances

The theorem of Kolmogorov stating that a non-negative definite kernel on ${N^1} \times {N^{ - 1}}$ is the covariance of a stochastic process on ${N^1}$ is generalized to continuous nonnegative definite functions on $Y \times Y,Y$ being a separable Hausdorff space. Also, a representation of such continuous nonnegative definite functions and their associated stochastic processes is provided.

On Functions Which are Superharmonic for Markov Processes

Let $X = (x_t ,\zeta ,\mathcal{M}_t ,{\bf P}_x )$ be a standard Markov process on a locally compact separable Hausdorff space $(E,\mathcal{O})$. An almost Borel measurable function $f(x):E \to ( { - \infty , + \infty } ]$ is called superharmonic if it satisfies the following conditions: a) it is intrinsically continuous; b) ${\bf M}_x f(x(\tau _G )) \leqq f(x)$ for any $x \in E$ and any open set G with a compact closure, where $\tau _G $ is the hitting time for the set $E - G$.The main results are stated in Theorems 1 and 2. In these theorems S denotes the set of $x \in E$ for which $x_t $ coincides with x (${\bf P}_x $ almost surely) during a positive random time interval $[0,\delta ]$; the symbol $\mathcal{U}$ denotes any open base of $\mathcal{O}$, and $\mathcal{V}$ is the class of all sets U of the type $U \in \mathcal{U}$ or $U = V - S$, where $V \in \mathcal{U}$.Theorem 1.A non-negative almost Borel function$f(x)$, $x \in E$, is superharmonic if and only if it is intrinsically continuous and\[ {\bf ...

Paracompactness and an example due to F. B. Jones

At the summer meeting (1955) of the American Mathematical Society, Mary E. Rudin presented an example of a separable normal nonparacompact space. It is the purpose of this note to point out that an example [3] due to F. B. Jones (1937) with an obvious definition of open sets is also such an example. Jones' paper was published before the notion of paracompactness appeared in the literature. The reader is referred to [1; 2] for definitions concerning paracompactness. EXAMPLE (JONES). Let M1 denote a subset of the open number interval I(0, 1) of cardinality K, such that each countable subset of M1 is an inner limiting set with respect to M1. Let Z1 denote the set of all points (x, y) of the number plane such that both x and y are positive rational numbers and 0 <x < 1. Furthermore, let a denote a most economical well ordered sequence of the points of M1, i.e., for p in M1, p is preceded in a by at most a countable subset of M1. Let S denote a space whose points are the points of M1 and Z1 in which open sets are defined as follows: (1) For p in Z1, p is an open set. (2) For p in M1 such that p has an immediate predecessor in a, an open set containing p is a point set D in M1+Z1 such that (a) DDp and (b) there exists an interior T of an inverted isosceles triangle with its lower vertex at p and whose base is parallel to the x-axis such that T. (M1+Z1) =D-p. (3) For p in M1 such that p has no immediate predecessor in a, let a denote a point of M1 such that a <p in a. Now, for a point x of Ml such that a <x <p in a, let T. denote the interior of an inverted isosceles triangle with its lower vertex at x and whose base is parallel to the x-axis. An open set D containing p is the set of all points y in S such that either y = x or y E T. * Zi. It is easy to see that S is a separable Hausdorff space. By a slight modification of Jones' argument, it may be shown that S is normal. In his argument where he considers K M1 to be countable, replace the set Dlk by a set Qlk such that Qlk = QM1 where Q is an open set in S such that (1) Q is countable, (2) QM1 is a closed subset of M1, and (3) Q.M1.H=QH=O.

The Intersection of Chains on a Topological Manifold

1. In previous papers, one of them in collaboration with S. Lefschetz,t the author has dealt with topological manifolds. A topological manifold, M1I,I, is a compact separable Hausdorff space (therefore metric) which has a complete set of neighborhoods each of which is a combinatorial n-cell (F. M., p. 393). The following properties are showln in F. M. and F. M. 2 to hold for ilI,: 1. the invariance of the homology characters; ? 2. the standard properties of the Kronecker Index of two chains on M1I. whose dimensions are p and l p; 3. the Poincare duality theorem. Property 1 was proved intrinsically, i. e. without imbedding M31,, in a Euclidean space of higher dimension and using the properties of the space residual to M,,. In 2, however, the imbedding space was used to prove that every non-bounding p-cycle oin M,, is cut by some (n p) -cycle oin MI with a Kronecker Index ? 1. From 2 follows 3. The present article makes no use of the imbedding theorem but defines intrinsically on ]II5 intersection cycles P17 (h = p + q n), for two chains, Cp and Cq, on ]II71 of dimensionality p and q, not meeting one another's boundaries; and proves intrinsically that the cycles thus obtained form a locally homologous family (L. T., p. 183) about the geometric intersection, G, (L. T., p. 182) of C. and Cq, thereby duplicating for 3M1, the salielnt theorem of the Lefschetz intersection theory for simplicial manifolds.