Abstract
Based on the natural vector addition and scalar multiplication, the set of all bounded and closed intervals in R cannot form a vector space. This is mainly because the zero element does not exist. In this paper, we endow a norm to the interval space in which the axioms are almost the same as the axioms of conventional norm by involving the concept of null set. Under this consideration, we shall propose two different concepts of open balls. Based on the open balls, we shall also propose the different types of open sets, which can generate many different topologies.
Highlights
In the real world, collecting data becomes an important issue for solving the practical and large-scaled problems in engineering, economics, and social sciences
In the financial market, stock price fluctuates violently, such that it may not be reasonable to record the price as an exact real number in a short period time
We can assume that the stock price is located in a bounded closed interval within a short period of time
Summary
In the real world, collecting data becomes an important issue for solving the practical and large-scaled problems in engineering, economics, and social sciences. Even though the interval space I cannot be a vector space, we still can endow a norm to the set of all bounded closed intervals by involving the concept of null set, and to study its topological structure. Based on the different types of openness, we shall study the topological structure of the normed interval space (I , k · k). The embedding theorem presented by Rådström [11] says that the family I consisting of all bounded closed intervals can be be embedded into a normed space isomorphically and isometrically. As the embedding approach of Rådström [11] is based on a specific normed space, N, defined above, the topological structures established in N is specific.
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