Determine the set of interior points, accumulation points, isolated points, and
ID: 2966789 • Letter: D
Question
Determine the set of interior points, accumulation points, isolated points, and boundary points for each of the following sets. Also Determine which of the following sets are open, which are closed, and which are neither open nor closed (please prove each statement)
(a) {1,1/2,1/3,1/4,1/5,...}
(b) {0}U{1,1/2,1/3,1/4,1/5,...}
(c) (0,1)U(1,2)U(2,3)U(3,4)UU(n,n+1)U...
(d) (1/2,1) U (1/4,1/2) U (1/8,1/4) U (1/16,1/8) U ...
(f) {x:x2 <2}
(g) RN
(h) RQ
Determine the set of interior points, accumulation points, isolated points, and boundary points for each of the following sets. Also Determine which of the following sets are open, which are closed, and which are neither open nor closed (please prove each statement) (a) {1,1/2,1/3,1/4,1/5,...} (b) {0}U{1,1/2,1/3,1/4,1/5,...} (c) (0,1)U(1,2)U(2,3)U(3,4)U U(n,n+1)U... (d) (1/2,1) U (1/4,1/2) U (1/8,1/4) U (1/16,1/8) U ... (e) {x:|x- (-infinite,0) U(0,infinite) Pi |Explanation / Answer
{1, 1/2, 1/3, 1/4, 1/5,....}
No interior points: this is right.
"The accumulation point is 0": right, but I would phrase it as "0 is the only accumulation point."
Every point is isolated: right.
"The boundary point is 0": same comment as above, but also this is wrong. The boundary of a set is its closure minus its interior. (The closure is the set itself plus the accumulation points.) So in this case, the closure is {1, 1/2, 1/3, ...}U{0}, the interior is the empty set, and the boundary is {1, 1/2, 1/3, ...}U{0}.
(0,1)U(1,2)U(2,3)U(3,4)U...U(n, n+1)U...
Every point is an interior point: right.
Every point is an accumulation point: right. But also, the points 0,1,2,3,... are accumulation points. The set of accumulation points is [0,infinity).
No points are isolated: right.
The boundary points are 0,1,2,3,... (or in set language, the boundary is {0,1,2,3,...}).
{x: x
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