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TopRs-Approximations In (Infinite) Universe
The approximation theory of RS is well known. For preciseness, we will recall the notion here. Let U be a classical set, called the universe. Let be a partition, namely, a family of subsets, called equivalence classes, that are mutually disjoint and their union is the whole universe U. Then the pair (U, ) is called approximation space in RS. Pawlak introduced following two definitions. Observe that Pawlak focus on finite universe. However we allow U to be infinite.
Let X be an arbitrary subset of the universe U.
Definition (RS) 1 Let E be an arbitrary equivalence class of R.
- 1.
Upper approximation:
- 2.
Lower approximation:
This definition is the formal form of the intuitive upper and lower approximations
Definition (RS)2 Let p be an arbitrary element of U.
In RS community, the previous definitions are directed generalized to Covering Cov by interpreting E as member of Cov .
TopCounter Intuitive Phenomena
In this section, we present some Counter Intuitive phenomena of approximations. The first example was generated to answer some questions raised in a conversation with Tian Yang, Guangming Lang, Jing Hao from Hunan University.
Example 1. Let the universe U be the real line. Let us consider the collection COV of all open half lines, namely, the sets of the following form {u | u < a} and {u | a < u} for a U. These half lines form a sub-base of the usual topology in real line; Here the “usual topology “ is a technical term referring to the topology of commonly known real line.