18.2.6. Problem. Let M be a metric space with the discrete metric. (a) Which sequences in...

18.2.6. Problem. Let M be a metric space with the discrete metric.

(a) Which sequences in M are Cauchy?

(b) Show that M is complete.

Expert Solution

Colby Messinger answered 2 years ago

Let (X, d) be a metric space. Prove that every metric space (X, d) is homeomorphic...

Let (X, d) be a metric space. Prove that every metric space (X, d) is homeomorphic to a metric space (Y, dY ) of diameter at most 1.

Prove that every sequence in a discrete metric space converges and is a Cauchy sequence. This...

Prove that every sequence in a discrete metric space converges and is a Cauchy sequence. This is all that was given to me... so I am unsure how I am supposed to prove it....

Show that id D is dense metric space of X and if all cauchy sequences Yn...

Show that id D is dense metric space of X and if all cauchy sequences Yn from a sense set D converge in D then X is complete.

Q2. Let (E, d) be a metric space, and let x ∈ E. We say that...

Q2. Let (E, d) be a metric space, and let x ∈ E. We say that x is isolated if the set {x} is open in E. (a) Suppose that there exists r > 0 such that Br(x) contains only finitely many points. Prove that x is isolated. (b) Let E be any set, and define a metric d on E by setting d(x, y) = 0 if x = y, and d(x, y) = 1 if x and y...

Let (X,d) be a metric space. The graph of f : X → R is the...

Let (X,d) be a metric space. The graph of f : X → R is the set {(x, y) E X X Rly = f(x)}. If X is connected and f is continuous, prove that the graph of f is also connected.

(a) Let <X, d> be a metric space and E ⊆ X. Show that E is...

(a) Let <X, d> be a metric space and E ⊆ X. Show that E is connected iff for all p, q ∈ E, there is a connected A ⊆ E with p, q ∈ E. b) Prove that every line segment between two points in R^k is connected, that is Ep,q = {tp + (1 − t)q | t ∈ [0, 1]} for any p not equal to q in R^k. C). Prove that every convex subset of R^k...

Let X be a metric space and t: X to X be a map that preserves...

Let X be a metric space and t: X to X be a map that preserves distances: d(t(x), t(y)) = d(x, y). Give an example in whicht is not bijective. Could let t: x to x+1,x non-negative, but how does this mean t is not surjective? Any help will be much appreciated!

Answer for a and be should be answered independently. Let (X,d) be a metric space, and...

Answer for a and be should be answered independently. Let (X,d) be a metric space, and a) let A ⊆ X. Let U be the set of isolated points of A. Prove that U is relatively open in A. b) let U and V be subsets of X. Prove that if U is both open and closed, and V is both open and closed, then U ∩ V is also both open and closed.

(Connected Spaces) (a) Let <X, d> be a metric space and E ⊆ X. Show that...

(Connected Spaces) (a) Let <X, d> be a metric space and E ⊆ X. Show that E is connected iff for all p, q ∈ E, there is a connected A ⊆ E with p, q ∈ E. b) Prove that every line segment between two points in R^k is connected, that is Ep,q = {tp + (1 − t)q | t ∈ [0, 1]} for any p not equal to q in R^k. C). Prove that every convex subset...

The metric space M is separable if it contains a countable dense subset. [Note the confusion...

The metric space M is separable if it contains a countable dense subset. [Note the confusion of language: “Separable” has nothing to do with “separation.”] (a) Prove that R^m is separable. (b) Prove that every compact metric space is separable.

Question