Examine the set S = {(x, y, z) R^3 | x^2 + y^2 + z^2 > 1, Q} to prove whether it

Question

Examine the set S = {(x, y, z) R^3 | x^2 + y^2 + z^2 > 1, Q} to prove whether it is open, closed or neither in R^3. In formulating your proof, you may create examples or counterexamples and you should also compare the meanings of the topological terms in this question. Examine the set T below to infer what its interior, closure and boundary are: T = {(x, y) R^2 | y > x^3} Let (M, || middot ||) be a normed space. Let x_n be a sequence in M such that x_n times x. Construct a proof that ||x_n|| rightarrow ||x|| in R. Consider the set X = {(x, y) R^2 ||x^2 + y^2] lessthanorequalto 3, |y| greaterthanorequalto 1}. Construct a proof to determine whether or not X is compact. Create an example of two subsets C, D R^2 and a point p R^2 such that CUD is disconnected but CUDU{p} is connected. In constructing your example, you should verify that you do indeed have a separation for C D

Explanation / Answer

c)if M is a normed linear space and Ysubset of X is a subspace then by restricting the norm toY

, we can consider Y as a normed linear space.

Clearly

| ||x||-||y|| less equal ||x||-||y||   and hence for any sequence {xn}subset of M

xn converges to x implies that ||xn|| converges to ||x|| in R

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