Interactive Real Analysis - part of MathCS.org

• Real Analysis
• 1. Sets and Relations
• 2. Infinity and Induction
• 3. Sequences of Numbers
• 4. Series of Numbers
• 5. Topology
• 5.1. Open and Closed Sets
• 5.2. Compact and Perfect Sets
• 5.3. Connected and Disconnected Sets
• 6. Limits, Continuity, and Differentiation
• 7. The Integral
• 8. Sequences of Functions
• 9. Historical Tidbits
• Java Tools

Examples 5.1.6(a):

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What is the boundary and the interior of (0, 4), [-1, 2], R, and O ? Which points are isolated and accumulation points, if any ?

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• The boundary of (0, 4) is the set consisting of the two elements {0, 4}. Every neighborhood of these two points contains points both from the interval (0,4) and from the complement of that interval. Therefore, both form the boundary. The interior of the set (0, 4) is the set (0, 4) (i.e. itself). No points of either set are isolated, and each point of the either set is an accumulation point. The same is true, incidentally for each of the sets (0, 4), [0, 4), (0, 4], and [0, 4].
• The boundary of [-1, 2] is the two-element set {-1, 2}, and the interior is (-1, 2). No points are isolated, and each point in either set is an accumulation point.
• The boundary of the set R as well as its interior is the set R itself. No point is isolated, all points are accumulation points.
• The boundary of the empty set as well as its interior is the empty set itself. Since the set contains no points, it can not contain isolated or accumulation points.