Closed set in a metric space

(See attached file for full problem description) If A is a closed set in a metric space (X,d) and , show that d(x,A)>0.

In a metric space (X,d) a closed ball with center x and radius r is defined to be the set

In a metric space (X,d) a closed ball with center x and radius r is defined to be the set . a. Show that B[x;r] is a closed set b. Give an example where

Open Neighborhood of A

(See attached file for full problem description) If A is a nonempty set in a metric space X and if r>0 show that is an open neighborhood of A.

Continuity

Show that any function from a discrete metric space X into a metric space Y is continuous.

Continuity

(See attached file for full problem description) Prove: The function f from the metric space X into the metric space Y is continuous if and only if is closed in X whenever F is closed in Y.

Show that R is homeomorphic to its subspace (0,1).

Show that R is homeomorphic to its subspace (0,1).

Continuity/closed

(See attached file for full problem description) Let f be a continuous real-valued function on a metric space X, , and . Show that E is a closed set.

Convergence = unique limit

Show that a convergent sequence in a metric space has a unique limit.

Taylor Polynomials : Finding nth Terms

Please see the attached file for the fully formatted problems.

Linear Regression Equations : 3 Equations, 3 Unknowns

How do you go mathamatically from eq. 12.4 to eq. 12.5 when solving three equations with three unknowns given the summation rules on page 460? Please show all steps of the mathamatical work by hand without using any computer programs