A NOTE ON THE MARTIN TOPOLOGY OF THE SPACE OF THE FORMAL BALLS
A NOTE ON THE MARTIN TOPOLOGY OF THE SPACE OF THE FORMAL BALLS
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关于正式球空间的马丁拓扑的注记
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发表时间：
2009
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通讯作者：
Hikari Hashiriura
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作者：
Hikari Hashiriura
设(X，d)是度量空间，BX=X×表示X中广义形式球的偏序集，我们研究了X的某些拓扑的Martin拓扑和乘积拓扑与Sorgenfrey线之间的关系。给出了Martin拓扑与度量拓扑和Sorgenfrey拓扑的乘积拓扑重合的条件，并考虑了Martin拓扑与度量拓扑和Sorgenfrey拓扑的乘积拓扑同胚的条件。我们还证明了具有Martin拓扑的形球空间与Sorgenfrey直线的平方是同胚的。1导论论域理论的拓扑学方法的基本工具是“类度量”函数，如拟度量和(弱)部分度量。Scott拓扑和Lawson拓扑是偏序集中与序结构相关的基本拓扑。几位作者研究了序结构、上述类度规函数、Scott拓扑和Lawson拓扑之间的关系。(4))。K.Martin(7)引入了域中度量的概念来描述程序上的定量语句，P.Waszkiewicz(10)讨论了由于S.G.Matthews(8)在连续偏序集上的度量与部分度量之间的关系。K.Martin还证明了每一次测量都会诱导出一种我们称之为Martin拓扑的拓扑。结果表明，Martin拓扑有一个开基，并且它比Lawson拓扑更强。然而，关于Martin拓扑，有几个事实是已知的。设(X，d)是度量空间。则B+X=X×(0，+∞)的元素称为形式球。我们在B+X上导出一个偏序为(x，r)�(y，S)i Fd(x，y)≤r−S。形式球的概念是由Wehrauch和Schreiber引入的，用来表示域中的度量空间作为计算模型(11)。几位作者将形式球的偏序集作为度量空间(1，2，5，6)的近似结构进行了研究。最近，Tsuiki-Hattori(9)引入了负半径的形式球，并研究了序关系类似于B+X的偏序集BX=X×R。BX的一个元素称为广义形式球。集合BX显然具有作为偏序集的Lawson拓扑。很容易看出，每个切片X×{t}⊂BX(t∈R)上的相对Lawson拓扑同胚于X的度量拓扑，而每个切片{x}×R⊂BX(x∈X)都同胚于通常的实直线R。在这个方向上，Tsuiki-Hattori考虑了BX上的Lawson拓扑与X和R的乘积拓扑的不同或重合。形式球空间上的马丁拓扑似乎更为复杂，因为每个切片X×{t}⊂BX(t∈R)上的相对马丁拓扑是离散空间，一般不与X的度量拓扑同胚，并且每个切片{x}×R⊂BX(x∈X)都是Sorgenfrey线的同胚。在本说明中，我们将考虑
Let (X,d) a metric space and BX = X × denote the partially ordered set of generalized formal balls in X. We investigate the relations between the Martin topology and the product topology of certain topologies of X and the Sorgenfrey line. We give a condition that the Martin topology coincides with the product topology of a metric topology and the Sorgenfrey topology, and consider on the conditions that the Martin topology is homeomorphic to the product topology of a metric topology and the Sorgenfrey topology. We also show that the space of formal balls on with the Martin topology is homeomorphic to the square of the Sorgenfrey lines. 1 Introduction Basic tools of topological approaches to domain theory are "metric-like" functions such as a qausi-metric and a (weak) partial metric. The Scott topology and the Lawson topology are known as the fundamental topologies related to the order structures in posets. Several authors investigated the relations between order structures, metric-like functions above, the Scott topology and the Lawson topology (cf. (4)). K. Martin (7) introduced a notion of a measurement in a domain to describe a quanti- tative statements on programs, and P. Waszkiewicz (10) discussed on the relations between the measurements and the partial metrics due to S. G. Matthews (8) on continuous posets. K. Martin also showed that every measurement induces a topology that we call the Martin topology. It is shown that the Martin topology has a clopen base, and it is stronger than the Lawson topology. However, a few facts are known about the Martin topology. Let (X, d) be a metric space. Then, an element of B + X = X ×(0, +∞) is called a formal ball. We induce a partial orderon B + X as (x, r) � (y, s )i fd(x, y) ≤ r − s. The notion of formal balls is introduced by Weihrauch and Schreiber to represent a metric space in a domain as a computational model (11). Several authors sudied the poset of formal balls as an approximating structure of a metric space (1, 2, 5, 6). Recently, Tsuiki-Hattori (9) introduced formal balls with negative radiuses and study the partially ordered set BX = X × R with an order relation which is similar to B + X. An element of BX is called a generalized formal ball. The sets BX obviously has the Lawson topology as a poset. It is easy to see that the relative Lawson topology on every slice X ×{ t }⊂ BX (t ∈ R) is homeomorphic to the metric topology of X and every slice {x }× R ⊂ BX (x ∈ X) is homeomorphic to the usual real line R. In this direction, Tsuiki- Hattori considered the differences, or coincidences of the Lawson topology and the product topology of X and R on BX. The Martin topology on the space of formal balls seems to be more complicated, because the relative Martin topology on every slice X ×{ t }⊂ BX (t ∈ R) is a discrete space, which is not homeomorphic to the metric topology of X in general, and every slice {x }× R ⊂ BX (x ∈ X) is homeomorphic to the Sorgenfrey line. In the present note, we will consider
DOI：
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发表时间：
2006
期刊：
影响因子：
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作者：
Y. Hattori;H. Tsuiki;M. Kimura;T. Imaoka and T. Iigai;K. Shoji;M. Kimura;M. Kimura;V. Chatyrko and Y. Hattori;V. Chatyrko and Y. Hattori;Y. Hattori and H. Tsuiki
通讯作者：
Y. Hattori and H. Tsuiki