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MAT371defs3 - 10 Theorem A set is compact iff it is...

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Chapter 3. 1. f : E R is continuous at x 0 if ... 2. x E is said to be an isolated point in E if there is an ² > 0 such that ( x - ², x + ² ) E = { x } . (Note that an isolated point of E is never an accumulation point of E . Also, a function is automatically continuous at isolated points in its domain.) 3. Uniform continuity . 4. Open set, closed set . 5. Open cover for a set. 6. The interior of a set, the closure of a set . 7. Compact set . 8. Sequentially compact set : We say that a set E is sequentially compact if every sequence { x n } n =1 of points from E has a subsequence { x n k } k =1 that converges to a point x E . 9. Heine Borel Theorem
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Unformatted text preview: . 10. Theorem. A set is compact iff it is sequentially compact. 11. Theorem. If E is a compact set and f : E → R is continuous, then f is uniformly continuous . 12. Theorem. If E is a compact set and f : E → R is continuous, then f ( E ) is compact . 13. The other theorems are useful, but you will not need to be able to reproduce them on an exam. However you will need to be able to use them when writing a proof. 14. Extreme Value Theorem . 15. Bolzano’s Theorem . 16. Intermediate Value Theorem ....
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