Abstract
In this paper, we have discussed the definitions and the basic properties of open sets, closed sets, accumulation points or limit points and sequence. Sets may be neither open nor closed. The reader should not confuse the concept “limit point of a set” with the different, though related, concept “limit of a sequence”. Some of the solved and supplementary problems will show the relationship between these two concepts. Observe that (an ∶n ∈ N) denotes a sequence and is a function. On the other hand, {an ∶n ∈ N} denotes the range of the sequence and is a set. We have given several characterizations of these sets. We discuss the definitions of bounded sequence, convergent sequence, Cauchy sequence and their relations. Then we can study every bounded sequence of real numbers contains a convergent subsequence and every Cauchy sequence of real numbers converges to a real number. Let A be a bounded, infinite set of real numbers. Then A has at least one accumulation point. We express the definition of topology, usual topology and topological space. The creation of topology the science of spaces and figures that remains unchanged under continuous deformations represents a phenomenon of this kind, but of a distinctly modern variety. Then we begin our study of some properties topological spaces by making the idea of being connected that is being in one piece. We observe that for the usual topology on the line R and in the plane R2. Finally, we express the characterizations of the discrete topological space and indiscrete topological space.
Highlights
INTRODUCTIONThe first one is the basis definitions of open sets and their properties on topology on the line R and in the plane R2
This paper provides the discussion in three sections
We study the basis properties of topological spaces
Summary
The first one is the basis definitions of open sets and their properties on topology on the line R and in the plane R2. (ii) The real line R, itself, is open since any open interval Sp must be a subset of R, that is p ∈ Sp ⊂ R. (v) The infinite open intervals, i.e., the subsets of R defined and denoted by (a, ∞) = *x|x ∈ R, x > a+, (−∞, a) = *x|x ∈ R, x < a+ and (−∞, ∞) = *x|x ∈ R+ = R are open sets. The infinite closed intervals, i.e., the subsets of R defined and denoted by ,a, ∞) = *x|x ∈ R, x ≥ a+, (−∞, a - = *x|x ∈ R, x ≤ a+ are not open sets since a ∈ R is not an interior point of either ,a, ∞) or International Journal of Advances in Scientific Research and Engineering (ijasre), Vol 5 (12), December-2019
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