The Topological Properties Of TVS-valuded Cone Metric Spaces

Metric spaces play an essential role in mathematical theory and applications, and lots of mathematical scholars have done plenty of work on it. Cone metric spaces, as one of the most important generalizations of classic metric spaces, have proven to be quite useful in nonlinear functional analysis, convex analysis, and optimization theory, etc.The first chapter brings forth the fundamental concepts of TVS-valued cone metric spaces and lists some recent developments in cone metric spaces, including some results on topology in TVS-valued cone metric spaces and the relevant fixed point theory. At first, we define the cone in real topological vector space, then a partial order in terms of this cone is shown, and according to it we come up with the idea of regular and normal cone. Finally, we get to the definition of cone metric spaces, in which convergent and Cauchy series, together with the concept of completeness are given, followed by a brief introduction to the fixed point theory of cone metric spaces in recent years.The main task of the second chapter is to construct the basic topological structure in TVS valued cone metric spaces. With the definition of the open ball brought forth based on a lemma, we conclude that the intersection of two open balls is the union of several other open balls, which indicates that similar to the case in metric spaces, all open balls form a topological base in TVS valued cone metric spaces. Then we define the open set in in TVS valued cone metric spaces through open balls, and that finish the construction of topology in TVS valued cone metric spaces.Based on chapter2, chapter3mainly deals with some important topological properties of TVS valued cone metric spaces, such as accountability and separation. Firstly, we prove that the TVS valued cone metric spaces satisfy the first axiom of accountability, then it is concluded a separable and Lindeloff TVS valued cone metric spaces satisfy the second axiom of accountability. Secondly, extending the concept of normal cone in Banach spaces to TVS, we define normal cone in locally convex spaces, and after proving some basic properties, we get the final result that locally convex space valued cone metric spaces are T3spaces.In chapter four, we discuss series in TVS valued cone metric spaces. Based on normal cones, we obtain some important results on convergent and Cauchy series, and the completeness properties of TVS valued cone metric spaces is discussed. Through what was discussed above, we generalized some famous theorems in metric spaces to TVS valued cone metric spaces.The boundedness and compactness in TVS valued cone metric spaces is fully discussed in chapter five. We get that in TVS valued cone metric spaces, countably compactness, sequentially compactness and compactness are equivalent, and based on such equivalence, we conclude that a TVS valued cone metric space is compact if and only if it is complete and totally bounded.Like in the case in metric spaces, we can also construct topological product of TVS valued cone metric spaces, which is the main topic in chapter six. We prove that the topological product of countability many complete locally convex space valued cone metric spaces is again a cone metric space, which is not the case for unaccountably many cone metric cone spaces. At last, the completeness and the boundedness of such topological product is given.In chapter seven, we construct an equivalent metric in TVS valued cone metric spaces, which makes a TVS valued cone metric space to be a metric space that is topological homeomorphism to the former TVS valued cone metric space, and that gives us a simplified proof of a theorem in cone metric space.