a) show that for all x in R and for any e > 0 that N(x; e) is open. b) show fro

Question

a) show that for all x in R and for any e > 0 that N(x; e) is open. b) show fro all x in R, and for any e > 0 N*(x; e) is open. a) show that for all x in R and for any e > 0 that N(x; e) is open. b) show fro all x in R, and for any e > 0 N*(x; e) is open.

Explanation / Answer

(a) to show a set A is open, we are required to show that every point a in A is an interior point of A. given x is a real number and e > 0 is (epsylon) then (x-e,x+e) is said to be the e( epsylon) neighbourhood of x. this can otherwise be denoted as N(x;e). suppose p is any point in N(x;e) choose e1= min { | p-e| , |p+e|} then clearly, (p-e1, p+e1) is a subset of N(x;e). therefore, p is an interior point of N(x;e). so, arbitrary point p of N(x;e) is an interior point of N(x,e). in other words, N(x,e) is an open set. -------------------------------------------------------------------------- the same argument as above is used to prove the second problem.

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