Prove the following: Let (Y, d) be a metric space. If Y is totally bounded, then Y is bounded.

Q: Prove C(x) with the supremum norm is a metric space by showing the supremum norm is a metric

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Q: Let (X, d) be a metric space, prove that for any x, y, z E X, we have |d(y, z) - d(x, z)| ≤d(x, y).

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Q: Bolzano-Weirstrass theorem for ℝ3 with the metric d((x1, x2, x3), (y1, y2, y3) = |x1 - y1| + |x2 -…

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Q: Let (X, d) be a metric space and let A, B⊆X be such that A is connected, and A∩B ≠ ∅ and A∩ (X − B)…

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Q: Let (X,d) be a metric space and E ⊆ X. Prove that if E is compact, then E is bounded

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Q: Prove that if f is a continuous mapping of a compact metric space, X, into a metric space Y, then…

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Q: Let (X,d) be a metric space. For x,y in X define e(x,y)=min{1,d(x,y)}. Prove that (X,e) is also a…

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Q: 2. Construct two metric spaces X and Y which are homeomorphic, but X is complete and Y is not…

A: 2. Construct two metric spaces  and  which are homeomorphic, but  is complete and  is not complete.

Q: Let (X, d) be a separable metric space. into l∞.

A: This is a problem of functional analysis.

Q: 5. Use the graph of f(x) below to find the following: lim f(x)= x-0 y=f(x) k. Where is f(x)…

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Q: For example (X,d) metric space with d is a Euclidean metric. if set A = { = |neN} CX nEN CX and…

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Q: Suppose (X, d) is a metric space and E a non-empty subset of X. Prove that E is disconnected if and…

A: Use definition of disconnectedness mentioned below.