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Unformatted text preview: NOTES ON POINT-SET TOPOLOGY These notes introduce some basic concepts in the field known as topology . Here we will only be concerned with point-set topology , essentially the topology of the set of real numbers. Topology is a huge area of mathematics, and you will see elements of it in some of your later courses; however, as before, our main purpose here is to see more examples of the types of proofs that come up in Math 104. We want to make sense of the notions of open and closed subsets of R . Intuitively, an open set should be one that “has no end”, and a closed set should be one that “ends”. The following definitions make this all precise. Definition 1. Let S ⊆ R . An element x of S is said to be an interior point of S if there exists > 0 such that the interval ( x- ,x + ) is contained in S . The interior of S , denoted by int( S ), is the set of all interior points of S . The interior of a set S is essentially the set of points in S that S “surrounds”. The above definition says that x ∈ S is in the interior of S if there is a small enough interval around x that is completely contained in S . Consider the interval [- 1 , 1]. Then 0 is an interior point of [- 1 , 1] since for = 1, the interval (0- , 0 + ) = (- 1 , 1) is contained in [- 1 , 1]. In fact, any point in (- 1 , 1) is an interior point of [- 1 , 1]. The point 1 ∈ [- 1 , 1], however, is not an interior point since there is no > 0 such that (1- , 1 + ) is contained in [- 1 , 1]; this is because any such interval will contain something larger than 1. It is easy to see that- 1 is also not an interior point of [- 1 , 1], so we conclude that int([- 1 , 1]) = (- 1 , 1). Definition 2. Let S ⊆ R . A real number x ∈ S is said to be a boundary point of S if for every > 0, the interval ( x- ,x + ) contains points in S and points not in S . The boundary of S , denoted by ∂S , is the set of all boundary points of S . The closure of S , denoted by S , is S ∪ ∂S . Intuitively, the boundary of a set is the set of points where S “stops” and goes no further. In the case of [- 1 , 1], 0 is not a boundary point since for = 1, the interval (0- , 0+ ) does not contain a point not in [- 1 , 1]. The point 1, however, is a boundary point since any interval around 1 will contain something in [...
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