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- Suppose (S,d) is a metric space. How can we prove that S is open?Show that close ball Y in a metfic space (X,d) is a closed set also show that if (X,d) is complete than (Y,d) is completeProve. For any x, y ∈ R and x < y, ⟨(x, y), ≤⟩ is not well-ordered, where (x, y) is an open interval and ≤ is the usual ordering.
- Prove that in a metric space (X, d) every closed ball that is a set K(x, r) = {y e X : d(x, y) <= r}, is closed set. Show on an example that closed ball K(x, r) does not have to be equal a closure of an open ball. signs on the imageLet (X, τ) be the topological space A,B⊂X. In this case, show that it is stated in the photo.Consider the collection of open sets, (-n,n) from n=1 to infinity in R with the usual metric d(x,y) = abs(x-y). Use this collection of open sets to show R is not compact (make sure to prove R = UneJ(-n,n) as part of your work.
- Suppose metric space X is not path-wise connected then X is not connected. True FalseLet (X, τ) be the topological space and A⊂X. In this case, show it as in the picture.Show that every plane through the origin in R³ may be identified with the null space of an element in (R³)*. State an analogous reult in R².
- Let X={a, b, c, d, e} and let T={X, non zero, {a}, {a, d}, {a, e}, then (X, T) is neither Hausdorff nor connected?let (x,t) be a topological space prove that (x,t) is not connected if and only if there exist A,B belongs to t with x= A union B and A intersect B = zeroare R is not connected if T is the indiscrete topology? Or if T is the trivial topology? Or if T is the finite closed topology?