Jiling Cao

Volterra spaces revisited

In this paper, we investigate Volterra spaces and relevant topological properties. New characterizations of weakly Volterra spaces are provided. An analogy of the Banach category theorem in terms of Volterra properties is obtained. It is shown that every weakly Volterra homogeneous space is Volterra, and there are metrizable Baire spaces whose hyperspaces of nonempty compact subsets endowed with the Vietoris topology are not weakly Volterra.

A Lattice-valued Banach-Stone Theorem

Let X and Y be compact Hausdorff spaces, and let E be a Banach lattice. In this short note, we show that if there exists a Riesz isomorphismΦ:C(X, E) →C(Y, R) such that Φ(f) has no zeros if f has none, then X is homeomorphic to Y and E is Riesz isomorphic to R.

Quasi-uniform hyperspaces of compact subsets

Abstract Let (X, u ) be a quasi-uniform space, K (X) be the family of all nonempty compact subsets of (X, u ) . In this paper, the notion of compact symmetry for (X, u ) is introduced, and relationships between the Bourbaki quasi-uniformity and the Vietoris topology on K(X) are examined. Furthermore we establish that for a compactly symmetric quasi-uniform space (X, u ) the Bourbaki quasi-uniformity u ∗ on K (X) is complete if and only if u is complete. This theorem generalizes the well-known Zenor-Morita theorem for uniformisable spaces to the quasi-uniform setting.

On generalized closed sets

Abstract In this paper we study generalized closed sets in the sense of N. Levine. We will consider the question of when some classes of generalized closed sets coincide. Also, some lower separation axioms weaker than T1 are investigated. We will provide characterizations of extremally disconnected spaces and sg-submaximal spaces by using various kinds of generalized closed sets.

Baire spaces and Vietoris hyperspaces

We prove that if the Vietoris hyperspace CL(X) of all nonempty closed subsets of a space X is Baire, then all finite powers of X must be Baire spaces. In particular, there exists a metrizable Baire space X whose Vietoris hyperspace CL(X) is not Baire. This settles an open problem of R. A. McCoy stated in 1975.

Some properties of quasiuniform multifunction spaces

The aim of this paper is to explore some properties of quasiuniform multifunction spaces. Various kinds of completeness of the quasiuniform multifunction space ( Y mX , U mX ) are characterized in terms of suitable properties of the range space ( Y, U ). We also discuss the local compactness of quasiuniform multifunction spaces. By using the notion of small-set symmetry, the classic result of Hunsaker and Naimpally is extended to the quasiuniform setting.

Pricing variance swaps under stochastic volatility and stochastic interest rate

In this paper, we investigate the effects of imposing stochastic interest rate driven by the Cox-Ingersoll-Ross process along with the Heston stochastic volatility model for pricing variance swaps with discrete sampling times. A dimension reduction mechanism based on the framework of Little and Pant (2001) is applied which later reduces to solving two three-dimensional partial differential equations. A semi-closed form solution to the fair delivery price of a variance swap is obtained via the derivation of characteristic functions. Practical implementation of this hybrid model is demonstrated through numerical simulations.

Amsterdam properties of Wijsman hyperspaces

In this paper we show the following results: (i) there exists a separable metric space of the first category whose Wijsman hyperspace is almost countably subcompact; (ii) there exists a σ-discrete crowded metric space whose Wijsman hyperspace is countably base-compact. Neither of these can occur with Vietoris hyperspaces.

Baire spaces, Tychonoff powers and the Vietoris topology

In this paper, we show that if the Tychonoff power -Yω of a quasi-regular space X is Baire, then its Vietoris hyperspace 2 X is also Baire. We also provide two examples to show (i) the converse of this result does not hold in general, and (ii) the Baireness of finite powers of a space is insufficient to guarantee the Baireness of its hyperspace.

Topological properties defined by games and their applications

Abstract In this paper, we study some two person games and some topological properties defined by them. As an application of our results we refine the Choquet–Dolecki theorem on multifunctions and the classical Vainstein lemma on closed maps.