Is N: a. Open and does it have interior points? b. Closed and what are its limit points? c. Bounded? d. Compact? e. Perfect? f. Open cover/finite subcover?

1. Exhibit an open cover of the interval (1, 2] that has no finite subcover.

For the subset "The set of points on the line given by x+2y=0" verify whether it is a sub space of R2

Please answer with explanations

Let S = {(x,y) R²: (x-1)² + y² ≤ 4 and y> √3}. Question a. Find the interior, exterior and boundary of the set S. b. Determine whether the set S is open, closed, or neither. Explain your answer. c. Determine whether or not the set S is bounded. Justify your answer.

Find an open cover of the interval (−∞, 3]. Justify why that open cover has no finite subcover.

3. Let S = {x1, x2, . . . , xn} be a nonempty, finite set of points.a. What is the interior of S?b. What is the boundary of S?c. Is S open, closed, or neither?All of the notions introduced in this lecture can be applied to sets of realnumbers: write x for x and |x| for k x k. A neighborhood of a number x0 isthen a set of the form{x : |x − x0| < δ} with δ > 0