I need help with #4. 1. Let E and F be compact sets in a metric space (X, d). Show that EUF is compactusing the definition of compactness.2. Let (X, d) be a metric space and EC X. Prove that if E is compact, then E isbounded.3. Suppose E C R. With respect to the Euclidean metric, is it possible that E':neN. If so, give an example; otherwise explain.4. Let (X, d) be a metric space, r E X, e > 0, and E = {y E X : d(x, y) < e}. Show thatE is closed.d(r, y)Show1+ d(x,y)'5. Let (X, d) be a metric space. Define f : X x X → R by f(r, y) =that f is a metric on X.

Let (X,d) be a metric space, with x∈X , ε>0, and E={y∈X:d(x,y)≤ε} To show: E is closed. That is…

Answered: 4. Let (X, d) be a metric space, r E X,…

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4. Let (X, d) be a metric space, r E X, e > 0, and E = {y E X : d(x, y) < e}. Show that E is closed.

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Question

I need help with #4.

1. Let E and F be compact sets in a metric space (X, d). Show that EUF is compact
using the definition of compactness.
2. Let (X, d) be a metric space and EC X. Prove that if E is compact, then E is
bounded.
3. Suppose E C R. With respect to the Euclidean metric, is it possible that E'
:neN. If so, give an example; otherwise explain.
4. Let (X, d) be a metric space, r E X, e > 0, and E = {y E X : d(x, y) < e}. Show that
E is closed.
d(r, y)
Show
1+ d(x,y)'
5. Let (X, d) be a metric space. Define f : X x X → R by f(r, y) =
that f is a metric on X.

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