interior point of S if there exists a neighborhood N x for some such that N x

# Interior point of s if there exists a neighborhood n

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This preview shows page 3 - 6 out of 10 pages. 11. Let S = Q . Then int S = , and bd S = R . 12. Think about a cell in the space R 3 , its membrane is the “boundary” of the cell. For a region in R 2 or a solid region in R 3 , we can observe its boundary. The above definition can be used for these cases. In general, the boundary of a region could be very very complicated, but it all can be defined by the above definition. Open sets and closed sets Let S R . Definition 1. S is open if every point x S is an interior point of S . 2. S is closed if bd S S . [Examples and remarks] 1. The basic model for “open sets” is an open interval ( a, b ). This may be why we use the name “open.” The basic model for “closed set” is a closed interval [ a, b ]. This may be why we use the name “closed.” 2. The most significance of the notion of “open sets” is that it can be used to define “continuous functions” between general sets. For example, for a function f : R R , we can define that f is continuous ⇐⇒ f - 1 ( U ) is an open set, for any open set U . (1) where f - 1 ( U ) = { x R | f ( x ) U } . 3. For any finite numbers a and b , ( a, b ] and [ a, b ) are neither open nor closed. 4. Let S = {- 1 } ∪ (0 , 2) is neither open nor closed. 5. The set of a finite number of points S = { a 1 , a 2 , ..., a n } is closed. 6. ( a, ) is open, and [ a, ) is closed. 7. ( -∞ , b ) is open, and ( -∞ , b ] is closed. 8. Let S = { 1 , 1 2 , 1 3 , ..., 1 n , ... } . Then S is either open, nor closed. 9. Let S = N . Then S is not open but is closed. 34   • • • 