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.

Expert Solution

Colby Messinger answered 2 years ago

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...

(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...

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.

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...

1(i) Show, if (X, d) is a metric space, then d∗ : X × X →...

1(i) Show, if (X, d) is a metric space, then d∗ : X × X → [0,∞) defined by d∗(x, y) = d(x, y) /1 + d(x, y) is a metric on X. Feel free to use the fact: if a, b are nonnegative real numbers and a ≤ b, then a/1+a ≤ b/1+b . 1(ii) Suppose A ⊂ B ⊂ R. Show, if A is not empty and B is bounded below, then both inf(A) and inf(B) exist and...

b）Prove that every metric space is a topological space. (c) Is the converse of part (b)...

b）Prove that every metric space is a topological space. (c) Is the converse of part (b) true? That is, is every topological space a metric space? Justify your answer

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....

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!

A metric space X is said to be locally path-connected if for every x ∈ X...

A metric space X is said to be locally path-connected if for every x ∈ X and every open neighborhood V of x in X, there exists a path-connected open neighborhood U of x in X with x ∈ U ⊂ V. (a) Show that connectedness + local path-connectedness ⇒ path-connectedness (b) Determine whether path-connectedness ⇒ local path-connectedness.

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